Vocabulary reduction identity

Key Concept 1

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

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

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

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

Example 2 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 255tOriginal 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 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 ( )t = 1 = 4[sin(210)cos(45) + cos(210)sin(45)]Sine Sum Identity

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

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

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

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

Example 4 Write as an Algebraic Expression Answer:

Example 5 Verify Cofunction Identities Verify cos (–θ) = cos θ. cos (–θ)= cos (0 – θ) Rewrite as a difference. = cos 0 cos θ + sin 0 sin θCosine Difference Identity = 1 cos θ + 0 sin θcos 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 6 Verify Reduction Identities Simplify. A. Verify. Cosine Difference Identity

Example 6 Verify Reduction Identities Answer:

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

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

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

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