reduction identity Evaluate a Trigonometric Expression A Find

• reduction identity

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

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

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.

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

Evaluate a Trigonometric Expression Multiply. Simplify. Answer:

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 255 t, where 255 is a degree measure. Rewrite the formula in terms of the sum of two angle measures. i = 4 sin 255 t = 4 sin (210 t + 45 t) Original equation 255 t = 210 t + 45 t The formula is i = 4 sin (210 t + 45 t). Answer: i = 4 sin (210 t + 45 t)

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 255 t. Use a sum identity to find the exact current after 1 second. i = 4 sin (210 t + 45 t) Rewritten equation = 4 sin (210 + 45) t=1 = 4[sin(210)cos(45) + cos(210)sin(45)] Sine Sum Identity

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

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

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

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

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.

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 .

Write as an Algebraic Expression Now apply substitution and simplify.

Write as an Algebraic Expression Answer:

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 = 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

Verify Reduction Identities A. Verify . Cosine Difference Identity Simplify.

Verify Reduction Identities Answer:

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

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

Solve a Trigonometric Equation Simplify. Divide each side by 2. Substitute. Solve for cos 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).