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Five-Minute Check (over Lesson 5-3) Then/Now New Vocabulary Key Concept: Sum and Difference Identities Example 1: Evaluate a Trigonometric Expression Example 2: Real-World Example: Use a Sum or Difference Identity Example 3: Rewrite as a Single Trigonometric Expression Example 4: Write as an Algebraic Expression Example 5: Verify Cofunction Identities Example 6: Verify Reduction Identities Example 7: Solve a Trigonometric Equation Lesson Menu

Solve for all values of x. B. C. D. 5–Minute Check 1

Find all solutions of 2cos2 x + 3cos x + 1 = 0 in the interval [0, 2π). B. C. D. 5–Minute Check 2

Find all solutions of 4 cos2 x = 5 – 4 sin x in the interval [0, 2π). B. C. D. 5–Minute Check 3

Find all solutions of sin x + cos x = 1 in the interval [0, 2π). B. C. D. 5–Minute Check 4

Solve 4 sin θ – 1 = 2 sin θ for all values of θ. B. C. D. 5–Minute Check 5

Use sum and difference identities to evaluate trigonometric functions. You found values of trigonometric functions using the unit circle. (Lesson 4-3) Use sum and difference identities to evaluate trigonometric functions. Use sum and difference identities to solve trigonometric equations. Then/Now

reduction identity Vocabulary

Key Concept 1

A. Find the exact value of cos 75°. Evaluate a Trigonometric Expression A. Find the exact value of cos 75°. 30° + 45° = 75° Cosine Sum Identity Example 1

Multiply. Combine the fractions. Answer: Evaluate a Trigonometric Expression Multiply. Combine the fractions. Answer: Example 1

B. Find the exact value of tan . Evaluate a Trigonometric Expression B. Find the exact value of tan . Write as the sum or difference of angle measures with tangents that you know. Example 1

Rationalize the denominator. Evaluate a Trigonometric Expression Tangent Sum Identity Simplify. Rationalize the denominator. Example 1

Multiply. Simplify. Simplify. Answer: Evaluate a Trigonometric Expression Multiply. Simplify. Simplify. Answer: Example 1

Find the exact value of tan 15°. B. C. D. Example 1

Rewrite the formula in terms of the sum of two angle measures. Use a Sum or Difference Identity A. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t, where 255 is a degree measure. Rewrite the formula in terms of the sum of two angle measures. Rewrite the formula in terms of the sum of two angle measures. i = 4 sin 255t Original equation = 4 sin (210t + 45t) 255t = 210t + 45t The formula is i = 4 sin (210t + 45t). Answer: i = 4 sin (210t + 45t) Example 2

Use a sum identity to find the exact current after 1 second. Use a Sum or Difference Identity B. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 255t. Use a sum identity to find the exact current after 1 second. Use a sum identity to find the exact current after 1 second. i = 4 sin (210t + 45t) Rewritten equation = 4 sin (210 + 45) t = 1 = 4[sin(210)cos(45) + cos(210)sin(45)] Sine Sum Identity Example 2

The exact current after 1 second is amperes. Use a Sum or Difference Identity Substitute. Multiply. Simplify. The exact current after 1 second is amperes. Answer: amperes Example 2

A. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 210t, where 210 is a degree measure. Rewrite the formula in terms of the sum of two angle measures. A. i = 4 sin (240t – 30t) B. i = 4 sin (180 + 30) C. i = 4 sin [7(30t)] D. i = 4 sin (150t + 60t) Example 2

B. ELECTRICITY An alternating current i in amperes in a certain circuit can be found after t seconds using i = 4 sin 210t, where 210 is a degree measure. Use a sum identity to find the exact current after 1 second. A. –1 ampere B. –2 amperes C. 1 ampere D. 2 amperes Example 2

A. Find the exact value of Rewrite as a Single Trigonometric Expression A. Find the exact value of Tangent Difference Identity Simplify. Substitute. Answer: Example 3

Rewrite as fractions with a common denominator. Rewrite as a Single Trigonometric Expression B. Simplify Sine Sum Identity Rewrite as fractions with a common denominator. Simplify. Answer: Example 3

Find the exact value of . A. 0 B. C. D. 1 Example 3

Applying the Cosine Sum Identity, we find that Write as an Algebraic Expression Write as an algebraic expression of x that does not involve trigonometric functions. Applying the Cosine Sum Identity, we find that Example 4

Write as an Algebraic Expression If we let α = and β = arccos x, then sin α = and cos β = x. Sketch one right triangle with an acute angle α and another with an acute angle β. Label the sides such that sin α = and cos β = x. Then use the Pythagorean Theorem to express the length of each third side. Example 4

Write as an Algebraic Expression Using these triangles, we find that = cos α or , cos (arccos x) = cos β or x, = sin α or , and sin (arccos x) = sin β or . Example 4

Now apply substitution and simplify. Write as an Algebraic Expression Now apply substitution and simplify. Example 4

Write as an Algebraic Expression Answer: Example 4

Write sin(arccos 2x + arcsin x) as an algebraic expression of x does not involve trigonometric functions. A. B. C. D. Example 4

cos (–θ) = cos (0 – θ) Rewrite as a difference. Verify Cofunction Identities Verify cos (–θ) = cos θ. cos (–θ) = cos (0 – θ) Rewrite as a difference. = cos 0 cos θ + sin 0 sin θ Cosine Difference Identity = 1 cos θ + 0 sin θ cos 0 = 1 and sin 0 = 0 = cos θ  Multiply. Answer: cos (–θ) = cos (0 – θ) = cos 0 cos θ + sin 0 sin θ = 1 cos θ + 0 sin θ = cos θ Example 5

Verify tan (– ) = –tan . A. B. C. D. Example 5

Cosine Difference Identity Verify Reduction Identities A. Verify . Cosine Difference Identity Simplify.  Example 6

Verify Reduction Identities Answer: Example 6

B. Verify tan (x – 360°) = tan x. Verify Reduction Identities B. Verify tan (x – 360°) = tan x. Tangent Difference Identity tan 360° = 0 Simplify.  Answer: Example 6

Verify the cofunction identity . A. B. C. D. Example 6

Find the solutions of on the interval [ 0, 2). Solve a Trigonometric Equation Find the solutions of on the interval [ 0, 2). Original equation Sine Sum Identity and Sine Difference Identity Example 7

Simplify. Divide each side by 2. Substitute. Solve for cos x. Solve a Trigonometric Equation Simplify. Divide each side by 2. Substitute. Solve for cos x. Example 7

On the interval [0, 2π), cos x = 0 when x = Solve a Trigonometric Equation On the interval [0, 2π), cos x = 0 when x = Answer: CHECK The graph of has zeros at on the interval [ 0, 2π).  Example 7

Find the solutions of on the interval [0, 2π). B. C. D. Example 7

End of the Lesson