6. Trigonometric Identities and More Equations

Sum and Difference Identities

Open Question
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = cos² x - sin² x
Open Question
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = 1 - 2 sin² x
Open Question
Verify that each equation is an identity (Hint: cos 2x = cos(x + x).)
cos 2x = (cot² x - 1)/(cot² x + 1)
Open Question
Match each expression in Column I with its equivalent expression in Column II.
(tan (π/3) - tan (π/4))/(1 + tan (π/3) tan (π/4))
Open Question
Be sure that you've familiarized yourself with the first set of formulas presented in this section by working C1–C4 in the Concept and Vocabulary Check. In Exercises 1–8, use the appropriate formula to express each product as a sum or difference.3x xcos -------- sin -------2 2
Open Question
Use one or more of the six sum and difference identities to solve Exercises 13–54.In Exercises 13–24, find the exact value of each expression.sin(60° - 45°)
Open Question
In Exercises 14–19, use a sum or difference formula to find the exact value of each expression.sin 195°
Open Question
Use one or more of the six sum and difference identities to solve Exercises 13–54.In Exercises 13–24, find the exact value of each expression.sin 75°
Open Question
In Exercises 14–19, use a sum or difference formula to find the exact value of each expression.5𝝅tan ---------12
Open Question
Use one or more of the six sum and difference identities to solve Exercises 13–54.In Exercises 13–24, find the exact value of each expression.( 𝝅 𝝅 )tan ( ------ + ------ )( 3 4 )
Open Question
Use one or more of the six sum and difference identities to solve Exercises 13–54.In Exercises 13–24, find the exact value of each expression.( 5𝝅 𝝅 )tan ( ------ ﹣ ------)( 3 4 )
Open Question
Use one or more of the six sum and difference identities to solve Exercises 13–54.In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.sin 40° cos 20° + cos 40° sin 20°
Open Question
Use one or more of the six sum and difference identities to solve Exercises 13–54.In Exercises 25–32, write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression.5𝝅 𝝅 5𝝅 𝝅sin ------- cos -------- ﹣ cos -------- sin -------12 4 12 4
Open Question
In Exercises 35–38, find the exact value of the following under the given conditions:d. sin 2α3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2
Open Question
In Exercises 35–38, find the exact value of the following under the given conditions:c. tan(α + β)3 𝝅 12 𝝅 sin α = ------- , 0 < α < -------- , and sin β = --------- , --------- < β < 𝝅. 5 2 13 2