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Five-Minute Check (over Lesson 5-1) Then/Now New Vocabulary Example 1: Verify a Trigonometric Identity Example 2: Verify a Trigonometric Identity by Combining Fractions Example 3: Verify a Trigonometric Identity by Multiplying Example 4: Verify a Trigonometric Identity by Factoring Example 5: Verify an Identity by Working Each Side Separately Concept Summary: Strategies for Verifying Trigonometric Identities Example 6: Determine Whether an Equation is an Identity Lesson Menu

Find the value of the expression using the given information Find the value of the expression using the given information. If tan θ = , find cot θ. A. B. C. D. 5–Minute Check 1

Find the value of the expression using the given information Find the value of the expression using the given information. If sin θ = and cos θ = , find tan θ. A. B. C. D. 5–Minute Check 2

Find the value of the expression using the given information Find the value of the expression using the given information. If csc θ = 3 and cos θ < 0, find cos θ and tan θ. A. B. C. D. 5–Minute Check 3

Simplify csc x – csc x cos 2 x. A. sin x B. cos x C. csc x(1 + sin x) D. 1 – cos x 5–Minute Check 4

If sin θ = 0.59, find . A. −0.81 B. −0.59 C. 0.59 D. 0.81 5–Minute Check 5

You simplified trigonometric expressions. (Lesson 5-1) Verify trigonometric identities. Determine whether equations are identities. Then/Now

verify an identity Vocabulary

Verify a Trigonometric Identity Verify that . The left-hand side of this identity is more complicated, so transform that expression into the one on the right. Pythagorean Identity Reciprocal Identity Simplify.  Example 1

Verify a Trigonometric Identity Answer: Example 1

Verify that 2 – cos2 x = 1 + sin2 x. A. 2 – cos2x = –(sin2x + 1) + 2 = 1 + sin2x B. 2 – cos2x = 2 – (sin2x + 1) = 1 + sin2x C. 2 – cos2x = 2 – (1 + sin2x) + 2 = 1 + sin2x D. 2 – cos2x = 2 – (1 – sin2x) = 1 + sin2x Example 1

Start with the right hand side of the identity. Verify a Trigonometric Identity by Combining Fractions Verify that . The right-hand side of the identity is more complicated, so start there, rewriting each fraction using the common denominator 1 – cos2 x. Start with the right hand side of the identity. Common denominator Distributive Property Example 2

Divide out the common factor of sin x. Verify a Trigonometric Identity by Combining Fractions Simplify. Divide out the common factor of sin x. Simplify. Quotient Identity  Example 2

Answer: Verify a Trigonometric Identity by Combining Fractions Example 2

Verify that . A. B. C. D. Example 2

Verify a Trigonometric Identity by Multiplying Verify that . Because the left-hand side of this identity involves a fraction, it is slightly more complicated than the right side. So, start with the left side. Multiply the numerator and denominator by the conjugate of sec x – 1, which is sec x + 1. Multiply. Example 3

Multiply by the reciprocal of the denominator. Verify a Trigonometric Identity by Multiplying Pythagorean Identity Quotient Identity Multiply by the reciprocal of the denominator. Divide out the common factor of sin x. Example 3

Distributive Property Verify a Trigonometric Identity by Multiplying Distributive Property Rewrite the fraction as the sum of two fractions; Reciprocal Identity. Divide out the common factor of cos x. Example 3

Quotient Identity  Verify a Trigonometric Identity by Multiplying Example 3

Verify a Trigonometric Identity by Multiplying Answer: Example 3

Verify that . A. B. C. D. Example 3

Verify that cos x sec 2 x tan x – cos x tan3 x = sin x. Verify a Trigonometric Identity by Factoring Verify that cos x sec 2 x tan x – cos x tan3 x = sin x. cos x sec 2 x tan x – cos x tan3 x = cos x tan x (sec2 x – tan2 x) Start with the left-hand side of the identity. Factor. Pythagorean Identity = cos x tan x (1) Quotient Identity = Divide out the common factor of cos x. = sin x  Example 4

Verify a Trigonometric Identity by Factoring Answer: cos x sec 2 x tan x – cos x tan3 x = cos x tan x (sec2 x – tan2 x) = cos x tan x (1) = = sin x Example 4

Verify that csc x – cos x csc x – cos x cot x + cot x = sin x. B. C. D. Example 4

Write as the sum of two fractions. Verify an Identity by Working Each Side Separately Verify that . Both sides look complicated, but there is a clear first step for the expression on the left. So, start with the expression on the left. Write as the sum of two fractions. Simplify and apply a Reciprocal Identity. Example 5

Verify an Identity by Working Each Side Separately From here, it is unclear how to transform 1 + cot x into , so start with the right side and work to transform it into the intermediate form 1 + cot x. Pythagorean Identity Simplify. Factor. Example 5

Divide out the common factor of 1 – cot x. Verify an Identity by Working Each Side Separately Divide out the common factor of 1 – cot x. To complete the proof, work backward to connect the two parts of the proof. Write as the sum of two fractions. Simplify and apply a Reciprocal Identity. Multiply by . Example 5

Simplify. Pythagorean Identity Simplify.  Answer: Verify an Identity by Working Each Side Separately Simplify. Pythagorean Identity  Simplify. Answer: Example 5

Verify that tan2 x – sin2 x = sin2 x tan2 x. B. C. D. Example 5

Key Concept 6

Determine Whether an Equation is an Identity A. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. The equation appears to be an identity because the graphs of the related functions over [–2π, 2π] scl: π by [–1, 3] scl: 1 coincide. Verify this algebraically. Example 6

Divide out the common factor of sec x. Determine Whether an Equation is an Identity Pythagorean Identity Divide out the common factor of sec x. Example 6

Reciprocal Identities Determine Whether an Equation is an Identity Reciprocal Identities Simplify. Quotient Identity Answer: Example 6

Determine Whether an Equation is an Identity B. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find an x-value for which both sides are defined but not equal. Example 6

Determine Whether an Equation is an Identity The graphs of the related functions do not coincide for all values of x for which the both functions are defined. When , Y1 ≈ 1.43 but Y2 ≈ –0.5. The equation is not an identity. Example 6

Determine Whether an Equation is an Identity Answer: When , Y1 ≈ 1.43 but Y2 = –0.5. The equation is not an identity. Example 6

B. When , Y1 ≈ 0.71 but Y2 ≈ 0.29. The equation is not an identity. Use a graphing calculator to test whether is an identity. If it appears to be an identity, verify it. If not, find a value for which both sides are defined but not equal. A. The equation appears to be an identity because the graphs of the related functions over [–2π, 2π] scl: π by [–3, 3] scl: 1 coincide. B. When , Y1 ≈ 0.71 but Y2 ≈ 0.29. The equation is not an identity. Example 6

End of the Lesson