Math: Trigonometric Sum, Difference, and Double-Angle Identities Worksheet

Question 1

Use a sum identity to find the exact value of sin(75°).

Accepted Answer

Using sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°, sin(75°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Question 2

Use a difference identity to find the exact value of cos(15°).

Accepted Answer

Using cos(45° - 30°) = cos 45° cos 30° + sin 45° sin 30°, cos(15°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

Question 3

Use a sum or difference identity to find the exact value of tan(105°).

Accepted Answer

Using tan(60° + 45°) = (tan 60° + tan 45°)/(1 - tan 60° tan 45°), tan(105°) = (√3 + 1)/(1 - √3). Rationalizing gives tan(105°) = -2 - √3.

Question 4

If sin A = 3/5 and cos B = 12/13, with A and B both acute angles, find the exact value of sin(A + B).

Accepted Answer

Since A is acute and sin A = 3/5, cos A = 4/5. Since B is acute and cos B = 12/13, sin B = 5/13. Therefore, sin(A + B) = sin A cos B + cos A sin B = (3/5)(12/13) + (4/5)(5/13) = 56/65.

Question 5

If cos x = 5/13 and x is in Quadrant IV, find the exact value of sin(2x).

Accepted Answer

Since cos x = 5/13 in Quadrant IV, sin x = -12/13. Using sin(2x) = 2 sin x cos x, sin(2x) = 2(-12/13)(5/13) = -120/169.

Question 6

If sin θ = 7/25 and θ is in Quadrant II, find the exact value of cos(2θ).

Accepted Answer

Since sin θ = 7/25 in Quadrant II, cos θ = -24/25. Using cos(2θ) = cos²θ - sin²θ, cos(2θ) = (-24/25)² - (7/25)² = 576/625 - 49/625 = 527/625.

Question 7

Simplify the expression sin(x + y) + sin(x - y).

Accepted Answer

Using the sum and difference identities, sin(x + y) = sin x cos y + cos x sin y and sin(x - y) = sin x cos y - cos x sin y. Adding them gives 2 sin x cos y.

Question 8

Simplify the expression cos(a + b) + cos(a - b).

Accepted Answer

Using the sum and difference identities, cos(a + b) = cos a cos b - sin a sin b and cos(a - b) = cos a cos b + sin a sin b. Adding them gives 2 cos a cos b.

Question 9

Verify the identity cos(2x) = 1 - 2sin²x.

Accepted Answer

Starting with cos(2x) = cos²x - sin²x, replace cos²x with 1 - sin²x. This gives cos(2x) = (1 - sin²x) - sin²x = 1 - 2sin²x, so the identity is verified.

Question 10

Rewrite 2cos²x - 1 using a double-angle identity.

Accepted Answer

The expression 2cos²x - 1 is equal to cos(2x) by the double-angle identity for cosine.

Question 11

Solve for x on the interval 0° ≤ x < 360°: sin(2x) = 1.

Accepted Answer

The equation sin(2x) = 1 occurs when 2x = 90° + 360°k. Dividing by 2 gives x = 45° + 180°k. On 0° ≤ x < 360°, the solutions are x = 45° and x = 225°.

Question 12

Find the exact value of cos(2x) if tan x = 3/4 and x is in Quadrant I.

Accepted Answer

Using cos(2x) = (1 - tan²x)/(1 + tan²x), cos(2x) = (1 - (3/4)²)/(1 + (3/4)²) = (1 - 9/16)/(1 + 9/16) = (7/16)/(25/16) = 7/25.

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Math: Trigonometric Sum, Difference, and Double-Angle Identities

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