Algebra and Trigonometry (4th Edition)

Chapter 9

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all ; An identity is an equation that holds true ... more

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; An identity is an equation that holds true for ... more

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; Replace with , then simplify.  .  ; more

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; Replace with , then simplify.Write as . ; more

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; Replace with , then simplify.Express as... more

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  ; Replace with and with , then simplify.... more

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; Write as and as . Write all terms with a ... more

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; Replace with and with , then simplify. ; ... more

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; Replace with . Combine terms with a common ... more

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1 ; Apply the Pythagorean identity to write using... more

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; Replace with .Multiply both numerator and ... more

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; Replace with and with .Multiply both ... more

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1 ; Replace with and with . ; Multiply both ... more

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; Replace with and with , then simplify. ; ... more

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; Split the fraction into two parts. ; Replace ... more

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; Replace with and with . ; Multiply both ... more

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; Factor out . ; Apply the Pythagorean identity .... more

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; Factor out from .  ; Apply the Pythagorean ... more

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; Replace with .Multiply both numerator and ... more

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; Replace with and with .Multiply both ... more

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; Replace with . ; Multiply both numerator and ... more

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; Replace with . ; Multiply both numerator and ... more

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; Combine terms with common denominators.Expand ... more

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; Replace with .Combine terms with common ... more

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; Replace with and with .Multiply both ... more

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; Replace with and with . ; Multiply both ... more

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; Find the LCD, then convert each term into ... more

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; Combine terms with a common denominator. ; ... more

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; On the LHS, replace with , then cancel out ... more

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; On the LHS, replace with and with , then ... more

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; On the LHS, replace with and with , then ... more

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; On the LHS, replace with with with . ; Cancel ... more

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; On the LHS, replace with and with , then ... more

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; On the LHS,apply the Pythagorean identity to ... more

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; On the  LHS, replace by and by . ; Since , ... more

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; On the LHS, replaceby by by . ; Replace by . ... more

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; On the LHS, replace with and with $$\frac{\... more

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; On the LHS, expand using the property . ; ... more

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; Use the formula for the product of sum and ... more

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; On the LHS, replace with and with , then ... more

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; On the LHS, apply the Pythagorean identity to ... more

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; On the LHS, replace with . Combine terms using... more

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; Apply the Pythagorean identities to write ... more

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; On the LHS, apply the Pythagorean identities to... more

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; On the LHS, apply the Pythagorean identity to ... more

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; On the LHS, expand $$(\sin{x}+\cos{x})^2$$ ... more

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; On the LHS, replace with . Combine terms using... more

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; On the LHS, replace with .Combine terms using ... more

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; On the LHS, split the numerator and denominator... more

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; On the LHS, write as , then expand using the ... more

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; On the LHS, split the fraction into two parts.... more

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; On the LHS, replace with and with .Combine ... more

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; On the LHS, apply the Pythagorean identity to ... more

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; On the LHS, apply the Pythagorean identity to ... more

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; Split the expression on LHS into factors using ... more

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; On the LHS, apply the Pythagorean identities to... more

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; On the LHS, expand using the property . ; Apply... more

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; On the LHS, replacewith with with with . ; ... more

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; Start with the LHS.In the numerator, apply the ... more

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; On the LHS, replace with and with . ; ... more

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; On the LHS, replacewith with with with . ; ... more

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; On the LHS, replace with and with . ; ... more

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; On the LHS, combine terms using a common ... more

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; On the LHS, replacewith with with . ; Multiply ... more

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; On the LHS, replace with .Combine terms using ... more

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; On the LHS, factor using the formula for the ... more

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; On the LHS, replace with . ; Multiply both ... more

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; In the RHS, replace with and with . ; ... more

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; On the LHS, combine terms using a common ... more

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; On the LHS, split the fraction into two parts ... more

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; Start with the LHS.Combine terms using a common... more

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; On the LHS, replace with and with . ; ... more

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; On the LHS, factor the numerator using the ... more

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; On the LHS, factor the denominator using the ... more

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; On the LHS, multiply both numerator and  ... more

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; Start with the LHS.Multiply both numerator and ... more

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; Start with the LHS.Multiply both numerator and ... more

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; Start with the LHS. Replace with .Find the LCD... more

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; Start with the LHS. Multiply both numerator and... more

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; Start with the RHS.Multiply both numerator and ... more

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; Start with the LHS.Multiply both numerator and ... more

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; Start with the LHS.Replace with , then ... more

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; Start with the LHS.Multiply both numerator and ... more

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; Start with the LHS.Multiply both numerator and ... more

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; On the LHS, replace with and   with .Combine ... more

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; Start with the LHS.Replace with .Multiply both... more

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; Replace with in the given expression, then ... more

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; Replace with in the given expression, then ... more

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; Replace with in the given expression, then ... more

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; Replace with in the given expression, then ... more

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; Replace with in the given expression, then ... more

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; Substitute ; Square 5sec ; Factor out 25. ; ... more

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yes  ; Using a graphing facility, sketch the graph... more

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No ; Using a graphing facility, sketch the graph ... more

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No ; Using a graphing facility, sketch the graph ... more

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Yes ; Using a graphing facility, sketch the graph ... more

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; Apply the identity to simplify the LHS. ; ... more

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; Cross multiply. ; Subtract the terms that ... more

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; On the LHS, replace with .Write the first term... more

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; Replace with and with . ; Factor out from ... more

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; On the LHS, factor the numerator using the ... more

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; Write the LHS as .Factor it out using the ... more

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; Replace with , then simplify. ; Split the ... more

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; Replace with , then apply the law of ... more

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; Apply the Pythagorean identity to write using... more

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; Apply the law of exponents to split the ... more

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Yes ; Simplify the LHS.  Therefore, the given ... more

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No  ; Evaluate the LHS and RHS when . ; Compare ... more

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No ; Evaluate the LHS and RHS when . ; Compare the... more

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; On the LHS: replace with , with and with . ... more

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; On the LHS: replace with , with and with . ... more

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Yes. Since the graphs of  and overlap in the ... more

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Verify the identity . ; Start with as the LHS:... more

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means Let the side opposite angle as , opposite ... more

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Addition ; The Addition Formula for sine is stated... more

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Subtraction ; The Subtraction Formula for cosine ... more

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; Write as .Use the addition formula for Sine ... more

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; Write as .Use the subtraction formula for Sine... more

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; Write as .Use the addition formula for Cosine ... more

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; Write as .Use the addition formula for Cosine ... more

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; Write as .Use the subtraction formula for ... more

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; Write as .Use the addition formula for Tangent... more

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; Write as .Use the addition formula for Sine ... more

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; Write as .Use the addition formula for Cosine ... more

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; Write as .Use the subtraction formula for ... more

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; Write as .Use the subtraction formula for Sine... more

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; Write into .Use the addition formula for ... more

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  ; Write into .Use the addition formula for ... more

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  ; Rewrite the given expression as with and... more

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; Rewrite the given expression as with and... more

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; Rewrite the given expression as with and... more

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; Rewrite the given expression as with and... more

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  ; Rewrite the given expression as with and... more

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  ; Rewrite the given expression as with and... more

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; Express the left hand side (LHS) of the given ... more

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; Express the left hand side (LHS) of the given ... more

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  ; Express the left hand side (LHS) of the given ... more

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  ; Express the left hand side (LHS) of the given ... more

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; Expand the left hand side (LHS) of the given ... more

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  ;   ; Expand the left hand side (LHS) of the ... more

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; Expand the left hand side (LHS) of the equation... more

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; Expand the left hand side (LHS) of the equation... more

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; Express the left hand side (LHS) of the given ... more

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; Express the left hand side (LHS) of the given ... more

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Since and , by transitivity of equality. ; ... more

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; Expand the the first term of the left hand side... more

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; Expand the left hand side (LHS) of the given ... more

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; Expand the left hand side (LHS) of the equation... more

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; Expand the first term of the left hand side (... more

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; Expand the first term of the left hand side (... more

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; Express the left hand side (LHS) of the given ... more

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; Express the left hand side (LHS) of the given ... more

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; Express the left hand side (LHS) of the given ... more

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; Express the left hand side (LHS) of the given ... more

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; Express the left hand side (LHS) of the given ... more

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; Expand the left hand side (LHS) of the given ... more

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; Expand the left hand side (LHS) of the given ... more

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; Expand the left hand side (LHS) of the given ... more

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; Use associative property of addition to regroup... more

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  ; Expand the left hand side (LHS) of the given ... more

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  ; Write the given expression as where and .Use... more

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; Write the given expression as where and .Use ... more

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; Write the given expression as where and .Use ... more

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  ; Write the given expression as where and .Use... more

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; Find the sum of the expressions inside the ... more

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; Find the sum of the expressions inside the ... more

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; Begin by applying the subtraction formula for ... more

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; Begin by applying the subtraction formula for ... more

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; Write the subtraction formula for cosine, using... more

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; Write the subtraction formula for sine, using ... more

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; Write the addition formula for sine using the ... more

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; Write the addition formula for tangent, using ... more

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; Apply the formula for the sums of sines and ... more

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; Apply the formula for the sums of sines and ... more

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; Distribute 5 over the terms inside the ... more

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; Apply the formula for the sums of sines and ... more

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The left side of the equation can be written as . ... more

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The left side of the equation can be written as . ... more

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Let . Applying the cofunction identity, . Given ... more

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Label the points on the figure as follows: ; From ... more

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Let and . Substituting the expressions for and ... more

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Let and . Substituting the expressions for and ... more

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; Find .Apply the addition formula for tangent, .... more

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Applying the cofunction identity , where gives . ... more

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Using the definition of the tangent function, . ... more

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Double-Angle ; The formula which would find the ... more

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Half-Angle ; The formula which would find the ... more

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; Use the Pythagorean identity to find .Solve ... more

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; Apply the Pythagorean identity to find .Solve ... more

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; Apply the Pythagorean identity to find .Solve ... more

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; Use the reciprocal identity, .Solve for in the... more

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; Use the Pythagorean identity to find .Solve ... more

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; Use the reciprocal identity, .Solve for in the... more

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; Apply the Pythagorean identity to find .Solve ... more

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; Apply the Pythagorean identity to find .Solve ... more

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; Apply the product rule for exponents , to ... more

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; Apply the product rule for exponents , to ... more

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; Apply the product rule for exponents , to ... more

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; Apply the product rule for exponents , to ... more

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; Apply the product rule for exponents , to ... more

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  ; Write as .Apply the lowering of power formula... more

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; Replace   by , since is exactly . ; Apply the ... more

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; Replace by , since is exactly while is ... more

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; Replace by , since and are exactly . ; Apply... more

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; Replace   by , since is exactly . ; Apply the ... more

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; Replace by , since is exactly . ; Apply the ... more

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; Replace by , since is exactly . ; Apply the ... more

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; Replace by , since and are exactly . ; Apply... more

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; Replace by , since is exactly . ; Apply the ... more

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; Replace by , since is exactly . ; Apply the ... more

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; Replace by , since is exactly while is ... more

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; Replace by , since is exactly . ; Apply the ... more

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; Replace by , since is exactly . ; Apply the ... more

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; Write as , then apply the Addition Formula for... more

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; Write as , then apply the Addition Formula for... more

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,  , ; Write as .Apply the lowering of power ... more

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, , ; Multiply the given range for by to obtain... more

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, , ; Find using the reciprocal identity . ... more

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, , ; Since and is in Quadrant I, then must be... more

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, ,   ; Find using the Reciprocal identity . ... more

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,  ,  ; Find using the Reciprocal identity , by ... more

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; Replace by so that the expression becomes . ... more

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; Replace by so that the expression becomes .... more

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; Replace by so that the expression becomes .... more

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; Replace by so that the expression becomes . ... more

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; Replace by so that the expression becomes . ... more

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; Replace by so that the expression becomes .... more

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; Replace by so that the expression becomes .... more

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; Replace by so that the expression becomes .... more

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; Find using the Double-Angle Formula .  ... more

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; Draw a right triangle in Quadrant IV with in ... more

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; Draw a right triangle in Quadrant II with in ... more

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; Draw a right triangle in Quadrant I with in ... more

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; Apply the Product-to-Sum Formula with and to ... more

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; Apply the Product-to-Sum Formula with and to... more

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; Apply the Product-to-Sum Formula  with and ... more

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; Apply the Product-to-Sum Formula with and to... more

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; Write the given expression as a sum using the ... more

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; Write the given expression as a product using ... more

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; Write the given expression as a product using ... more

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; Write the given expression as a product using ... more

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  ; Write the given expression as a product using ... more

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; Write the given expression as a product using ... more

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; Write the given expression as a product using ... more

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; Write the given expression as a difference ... more

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; Write the given expression as a sum using the ... more

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; Write the given expression as a difference ... more

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; Write the given expression as a product using ... more

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; Write the given expression as a product using ... more

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; Write the given expression as a product using ... more

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; Write the LHS using by applying the Double-... more

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; Write the RHS using by applying the Double-... more

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; Expand the LHS. ; Apply the Pythagorean ... more

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; Factor the LHS using the property with and... more

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; On the LHS, replace with and with . ; ... more

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; Simplify the LHS using the Half-Angle Formula ... more

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  ; Factor out from the LHS ; Apply the Half-... more

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; Start with the LHS.Apply the Half-Angle Formula... more

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  ; On the LHS, write as , then apply the Double-... more

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; Split the LHS into two terms, with as one term... more

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; On the LHS,  factor the denominator using the ... more

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; Simplify the RHS using the Half-Angle Formula ... more

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; Write using by applying the Double-Angle ... more

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; Write the LHS as sum of cubes where and , ... more

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; On the LHS, write as , then apply the Addition... more

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; Start with the LHS.Write as and as .Expand ... more

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; Start with the LHS.W as a product using the Sum... more

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; Start with the LHS.Write the numerator as a ... more

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; Start with the LHS.Write the numerator as , ... more

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; Start with the LHS.On the numerator, write as ... more

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; Start with the LHS.Write the numerator as a ... more

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; On the RHS, multiply both numerator and ... more

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; On the LHS, write using and , by applying the... more

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; Start with the LHS.Write the second factor as... more

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; Write the LHS as a product using the Sum-to-... more

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; Write the LHS as a product using the Sum-to-... more

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; Write the LHS as a product using the Sum-to-... more

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; Write the LHS as a product using the Sum-to-... more

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; Start with the LHS.Regroup the terms in the ... more

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; On the LHS, substitute in , then express it ... more

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; On the LHS, replace by , so that Write the ... more

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; On the LHS, replace by , so that .Since , draw... more

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From example (2), .Substitute in the given ... more

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Consider the figure,  Let the area of the triangle... more

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Use the sum of angle formula, in the first two ... more

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Consider the triangle OAB and find the value of... more

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Consider the triangle . ;  Find the area of the ... more

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infinitely many ; The trigonometric functions are ... more

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no ; The range of is from to , so has no ... more

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0.3;   ; Since and , so . From the graph, two ... more

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,   ; Find the angles.Take the inverse sine.... more

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,   ; Find the angles.Take the inverse sine.... more

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; Find the angles.Take the inverse cosine.This ... more

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,   ; Find the angles.Take the inverse cosine.... more

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, ; Find the angles.Take the inverse cosine.Use ... more

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, ; Find the angles.Take the inverse of sine.Use ... more

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, ; Find the angles.Take the inverse sine.Use the ... more

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, ; Find the angles.Take the inverse cosine.Use ... more

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; Find the angle.Take the inverse tangent. ; Find... more

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; Find the angle.Take the inverse tangent. ; Find... more

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; Find the angle.Take the inverse tangent.Use the... more

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; Find the angle.Take the inverse tangent.Use the... more

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; Find the angles.Take the inverse cosine.... more

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; Find the angles.Take the inverse of cosine.... more

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; Find the angles.Take the inverse sine.Subtract... more

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; Find the angles.Take the inverse sine.Subtract... more

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; Find the angles.Take the inverse of cosine.Use ... more

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; Find the angle.Take the inverse of tangent.Use ... more

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; Find the angle.Take the inverse of tangent.Use ... more

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; Find the angles.Take the inverse of sine.Use ... more

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; Find the angle.Subtract 1 from both sides of ... more

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; Find the angle.Subtract 1 from both sides of ... more

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,   ; Subtract 1 from both sides of the equation.... more

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,   ; Add 1 to both sides of the equation.Divide ... more

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, ; Add 1 to both sides of the equation.Divide ... more

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; Solve for .Subtract 1 from both sides.Divide ... more

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; Solve for .Add 1 on both sides.Divide both ... more

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; Solve for .Subtract 1 from both sides. ; Take ... more

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; Solve for .Add 1 on both sides.Divide both ... more

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; Solve for .Add 3 on both sides.Divide both ... more

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; Solve for .Add 4 on both sides. ; Solve for .... more

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; Solve for .Add 1 on both sides.Divide both ... more

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; Solve for .Add 2 on both sides. ; Solve for .... more

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; Solve for .Add 4 on both sides. ; Solve for .... more

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; Set each factor to 0. ; Solve for .Add 4 on ... more

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; Set each factor to 0. ; Solve for .Add 2 on ... more

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; Set the equation in form of .Factor the left ... more

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; Factor the left side of the equation. ; Set ... more

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; Factor the left side of the equation. ; Set ... more

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; Substitute . ; Factor the left side of the ... more

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; Factor the left side of the equation. ; Equate ... more

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; Factor the left side of the equation. ; Equate ... more

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No solution ; Factor the left side of the equation... more

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No solution ; Factor the left side of the equation... more

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; Subtract from both sides. ; Factor the left ... more

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; Subtract from both sides. ; Factor the left ... more

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; Equate each factor to 0. ; Solve for for . ; ... more

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; Equate each factor to 0. ; Solve for for . ; ... more

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; Factor the left side of the equation. ; Equate ... more

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; Factor the left side of the equation. ; Equate ... more

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; Factor the left side of the equation. ; Equate ... more

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; Factor the left side of the equation. ; Equate ... more

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; Substitute and . ; Solve for . ; more

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; Substitute and . ; Solve for . ; more

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; Apply . ; Solution of in are .General ... more

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; Apply .Subtract 1 from both sides.Multiply both... more

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No solution ; Apply .Subtract from both sides.... more

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; Apply .Subtract 1 from both sides.Multiply both... more

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; Apply .Subtract from both sides.Factor the ... more

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; Apply .Factor the left side of the equation. ; ... more

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; Apply .Factor the left side of the equation. ; ... more

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; Apply .Subtract from both sides.Multiply both ... more

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; Apply .Subtract from both sides. ; Solve for... more

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; Apply .Subtract 1 from both sides.Multiply both... more

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; Apply .Multiply both sides by . ; Solve for... more

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; Square both sides.Subtract from both sides.... more

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; Add on both sides.Square both sides.Subtract ... more

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; Square both sides.Apply .Subtract from both ... more

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,   ; Use the identity , and substitute it into ... more

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general solution: solutions in :   ; Solve for the... more

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general solution: solutions in : ; Solve for the ... more

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general solution: solutions in : ; Solve for the ... more

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general solution: solutions in : ; Solve for the ... more

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; Simplify the equation.Use the addition formula ... more

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; Simplify the equation.Use the subtraction ... more

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; Simplify the equation.Use the subtraction ... more

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; Simplify the equation.Use the subtraction ... more

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; Simplify the equation.Use the double angle ... more

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; Simplify the equation.Use the double angle ... more

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0 ; Simplify the equation.Use the double angle ... more

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; Simplify the equation.Use the following ... more

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; Simplify the equation.Use the double angle ... more

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; Simplify the equation and solve for .Use double... more

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; Simplify the equation.Write as .Use double-... more

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; Simplify the equation.Write as .Use double-... more

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