Question 1

If sin(35°) = k, then what is the value of cos(55°)?

Accepted Answer

Step 1: Identify that 35° and 55° are complementary angles because 35° + 55° = 90°. Step 2: Apply the complementary angle property: sin(θ) = cos(90° − θ). Step 3: Therefore, sin(35°) = cos(90° − 35°) = cos(55°). Step 4: Since sin(35°) = k, we have cos(55°) = k.

Question 2

If sin(35°) = cos(θ), where θ is an acute angle, find the value of θ.

Accepted Answer

Step 1: Recall the complementary angle property of trigonometric functions: sin(A) = cos(90° - A), or equivalently, sin(A) = cos(B) when A + B = 90°.

Step 2: Given sin(35°) = cos(θ), we apply the complementary angle relationship.

Step 3: Using sin(35°) = cos(90° - 35°), we get sin(35°) = cos(55°).

Step 4: Comparing sin(35°) = cos(θ) with sin(35°) = cos(55°), we have θ = 55°.

Therefore, answer is 55°.

Question 3

If sin(35°) = k, then what is the value of cos(55°)?

Accepted Answer

Step 1: Identify that 35° and 55° are complementary angles because 35° + 55° = 90°. Step 2: Apply the complementary angle property: sin(θ) = cos(90° − θ). Step 3: Therefore, sin(35°) = cos(90° − 35°) = cos(55°). Step 4: Since sin(35°) = k, we have cos(55°) = k.

Question 4

If sin(5θ + 12°) = cos(3θ + 28°), where both angles are acute, find the value of θ.

Accepted Answer

Step 1: Use the complementary angle property: sin(A) = cos(B) implies A + B = 90°. Step 2: Set up equation: (5θ + 12°) + (3θ + 28°) = 90°. Step 3: Simplify: 8θ + 40° = 90°. Step 4: Solve: 8θ = 50°, so θ = 6.25° = 25/4 degrees.

Question 5

If sin(5θ + 12°) = cos(3θ + 28°), where both angles are acute, find the value of θ.

Accepted Answer

Step 1: Use the complementary angle property: sin(A) = cos(B) when A + B = 90°. Step 2: Set up the equation: (5θ + 12°) + (3θ + 28°) = 90°. Step 3: Simplify: 8θ + 40° = 90°. Step 4: Solve for θ: 8θ = 50°, so θ = 6.25°. Therefore, the answer is 6.25°.

Question 6

If sin(3A − 15°) = cos(A + 25°), where 3A − 15° and A + 25° are both acute angles, find the value of A.

Accepted Answer

Step 1: Use the complementary angle identity: sin θ = cos(90° − θ). So sin(3A − 15°) = cos(90° − (3A − 15°)) = cos(105° − 3A). Step 2: Set equal to cos(A + 25°): cos(105° − 3A) = cos(A + 25°). Step 3: Since cosine is one-to-one for acute angles: 105° − 3A = A + 25°. Step 4: 105° − 25° = A + 3A → 80° = 4A → A = 20°. Therefore, the answer is 20°.

Question 7

If sin(5x + 20°) = cos(3x + 40°), where both angles are acute, then find the value of tan(2x + 15°).

Accepted Answer

Step 1: Use the complementary angle property: sin θ = cos(90° - θ). Therefore, sin(5x + 20°) = cos(90° - 5x - 20°) = cos(70° - 5x). Step 2: Set this equal to the given expression: cos(70° - 5x) = cos(3x + 40°). Step 3: Since both angles are acute and cosine is one-to-one in [0°, 90°], we have: 70° - 5x = 3x + 40°. Step 4: Solve for x: 70° - 40° = 3x + 5x → 30° = 8x → x = 3.75°. Step 5: Find 2x + 15° = 2(3.75°) + 15° = 7.5° + 15° = 22.5°. Step 6: Calculate tan(22.5°) = √2 - 1 (this is a standard value). Alternatively, tan(22.5°) can be computed using the half-angle formula: tan(45°/2) = (1 - cos 45°)/sin 45° = (1 - 1/√2)/(1/√2) = √2 - 1.

Question 8

Simplify: sin(48°)·sec(42°) + cos(48°)·csc(42°)

Accepted Answer

Step 1: Recognize that 48° and 42° are complementary angles (48° + 42° = 90°). Step 2: Apply complementary identities: sin(48°) = cos(42°) and cos(48°) = sin(42°). Step 3: Substitute: cos(42°)·sec(42°) + sin(42°)·csc(42°). Step 4: Use reciprocal identities: sec(42°) = 1/cos(42°) and csc(42°) = 1/sin(42°). Step 5: Simplify: cos(42°)·(1/cos(42°)) + sin(42°)·(1/sin(42°)) = 1 + 1 = 2.

Question 9

If cos(3x) = sin(2x), where both angles are acute, find the value of x.

Accepted Answer

Step 1: Use the complementary angle property: sin(θ) = cos(90° − θ). Step 2: Rewrite the equation as cos(3x) = cos(90° − 2x). Step 3: Since cosine is one-to-one for acute angles, equate the arguments: 3x = 90° − 2x. Step 4: Solve for x: 3x + 2x = 90°, so 5x = 90°. Step 5: Therefore, x = 18°.

Question 10

If cos(42°) = a, then sin(48°) equals:

Accepted Answer

Step 1: Recognize that 42° and 48° are complementary angles (42° + 48° = 90°). Step 2: Apply the complementary angle property: sin(θ) = cos(90° − θ). Step 3: Rewrite: sin(48°) = cos(90° − 48°) = cos(42°). Step 4: Since cos(42°) = a, we have sin(48°) = a.

Question 11

Given that sec(α) = 2, find the value of cosec(90° − α).

Accepted Answer

Step 1: From sec(α) = 2, we get cos(α) = 1/2. Step 2: Use the complementary angle property: cosec(90° − α) = sec(α). Step 3: Therefore, cosec(90° − α) = 2.

Question 12

If cos(θ) = sin(3θ), where θ is an acute angle, then θ equals:

Accepted Answer

Step 1: Use the complementary angle identity: cos(θ) = sin(90° − θ). Step 2: Set up the equation: sin(90° − θ) = sin(3θ). Step 3: Since both angles have the same sine value and θ is acute, the angles must be equal: 90° − θ = 3θ. Step 4: Solve: 90° = 4θ, so θ = 22.5°.

Question 13

What is sin(18°) × csc(72°) + cos(18°) × sec(72°)?

Accepted Answer

Step 1: Recognize that 18° and 72° are complementary angles (18° + 72° = 90°). Step 2: Apply complementary angle properties: sin(72°) = cos(18°) and cos(72°) = sin(18°). Step 3: Therefore, csc(72°) = 1/sin(72°) = 1/cos(18°) and sec(72°) = 1/cos(72°) = 1/sin(18°). Step 4: Substitute into the expression: sin(18°) × (1/cos(18°)) + cos(18°) × (1/sin(18°)). Step 5: Simplify: sin(18°)/cos(18°) + cos(18°)/sin(18°) = tan(18°) + cot(18°). Step 6: Express with common denominator: (sin²(18°) + cos²(18°))/(sin(18°)cos(18°)) = 1/(sin(18°)cos(18°)). Step 7: Using sin(2θ) = 2sin(θ)cos(θ), we get sin(18°)cos(18°) = sin(36°)/2. Step 8: Therefore, the answer is 2/sin(36°). For numerical purposes, sin(36°) ≈ 0.588, giving approximately 3.4. However, for a clean SSC answer, let me reconsider: tan(18°) + cot(18°) = 2/sin(36°). Since sin(36°) = √(10 − 2√5)/4, this doesn't yield a clean integer. Let me verify the setup is correct and provide the simplified form.

Question 14

Simplify: sec(37°) × sin(53°) − cos(37°) × csc(53°)

Accepted Answer

Step 1: Recognize that 37° and 53° are complementary angles (37° + 53° = 90°). Step 2: Apply complementary angle properties: sin(53°) = cos(37°) and csc(53°) = sec(37°). Step 3: Substitute into the expression: sec(37°) × cos(37°) − cos(37°) × sec(37°). Step 4: Simplify: sec(37°) × cos(37°) = 1 and cos(37°) × sec(37°) = 1. Step 5: Therefore, 1 − 1 = 0.

Question 15

Evaluate: tan(28°) × tan(62°) + sin²(40°) + sin²(50°)

Accepted Answer

Step 1: Recognize that 28° and 62° are complementary (28° + 62° = 90°), so tan(62°) = cot(28°) = 1/tan(28°). Step 2: Therefore, tan(28°) × tan(62°) = tan(28°) × cot(28°) = 1. Step 3: Recognize that 40° and 50° are complementary (40° + 50° = 90°), so sin(50°) = cos(40°). Step 4: Therefore, sin²(40°) + sin²(50°) = sin²(40°) + cos²(40°) = 1. Step 5: Final answer = 1 + 1 = 2.

Question 16

If tan(62°) = m, then cot(28°) equals:

Accepted Answer

Step 1: Identify that 62° and 28° are complementary angles (62° + 28° = 90°). Step 2: Apply the complementary angle identity: tan(θ) = cot(90° − θ). Step 3: Therefore, tan(62°) = cot(90° − 62°) = cot(28°). Step 4: Since tan(62°) = m, we have cot(28°) = m.

Question 17

If sin(3x) = cos(2x) and both angles are acute, what is the value of x in degrees?

Accepted Answer

Step 1: Use the complementary angle property: sin θ = cos(90° − θ). Step 2: Rewrite sin(3x) = cos(2x) as sin(3x) = sin(90° − 2x). Step 3: For acute angles, if sin A = sin B, then A = B. Step 4: Therefore, 3x = 90° − 2x. Step 5: Solve: 3x + 2x = 90°, so 5x = 90°, thus x = 18°.

Question 18

Given that cos(62°) = sin(x) and sin(28°) = cos(y), where x and y are acute angles, find x + y.

Accepted Answer

Step 1: Using the complementary angle property cos(A) = sin(90° - A), we have cos(62°) = sin(90° - 62°) = sin(28°). Therefore, sin(x) = sin(28°), which gives x = 28° (since x is acute). Step 2: Using sin(A) = cos(90° - A), we have sin(28°) = cos(90° - 28°) = cos(62°). Therefore, cos(y) = cos(62°), which gives y = 62° (since y is acute). Step 3: x + y = 28° + 62° = 90°.

Question 19

If cot(5x + 10°) = tan(3x + 20°), find the value of x.

Accepted Answer

Step 1: Use the complementary angle identity: cot(θ) = tan(90° − θ). Step 2: Rewrite: cot(5x + 10°) = tan(90° − (5x + 10°)) = tan(80° − 5x). Step 3: Set equal: tan(80° − 5x) = tan(3x + 20°). Step 4: For tan(A) = tan(B), we have A = B. Step 5: Therefore, 80° − 5x = 3x + 20°. Step 6: Solve: 80° − 20° = 3x + 5x, so 60° = 8x, thus x = 7.5°.

Question 20

Simplify: sin(42°) × sec(48°) + cos(42°) × csc(48°).

Accepted Answer

Step 1: Recognize that 42° and 48° are complementary angles (42° + 48° = 90°). Step 2: Apply complementary identities: sin(42°) = cos(48°) and cos(42°) = sin(48°). Step 3: Rewrite: sin(42°) × sec(48°) = cos(48°) × sec(48°) = cos(48°) × (1/cos(48°)) = 1. Step 4: Rewrite: cos(42°) × csc(48°) = sin(48°) × csc(48°) = sin(48°) × (1/sin(48°)) = 1. Step 5: Total = 1 + 1 = 2.

Question 21

If sin(3x) = cos(2x), where 0° < x < 30°, then what is the value of x?

Accepted Answer

Step 1: Use the complementary angle identity: sin(θ) = cos(90° − θ). Step 2: Rewrite the equation: sin(3x) = cos(2x) becomes sin(3x) = sin(90° − 2x). Step 3: For the equation sin(A) = sin(B), we have A = B (in the given range). Step 4: Therefore, 3x = 90° − 2x. Step 5: Solve: 3x + 2x = 90°, so 5x = 90°, thus x = 18°.

Question 22

Given that tan(28°) × tan(62°) = m, find the value of m.

Accepted Answer

Step 1: Observe that 28° and 62° are complementary angles (28° + 62° = 90°). Step 2: Apply the complementary angle identity: tan(θ) × tan(90° − θ) = tan(θ) × cot(θ) = 1. Step 3: Therefore, tan(28°) × tan(62°) = tan(28°) × cot(28°) = 1. Step 4: Thus, m = 1.

Question 23

If tan(3α) = cot(2α), where both angles are acute, then what is the value of α?

Accepted Answer

Step 1: Recognize that tan(θ) = cot(φ) implies θ and φ are complementary angles, i.e., θ + φ = 90°. Step 2: Apply this to the given equation: 3α + 2α = 90°. Step 3: Simplify: 5α = 90°. Step 4: Solve for α: α = 18°.

Question 24

Evaluate: [sin²(25°) + sin²(65°)] / [cos²(25°) + cos²(65°)].

Accepted Answer

Step 1: Recognize that 25° and 65° are complementary angles (25° + 65° = 90°). Step 2: Apply complementary identities: sin(65°) = cos(25°) and cos(65°) = sin(25°). Step 3: Rewrite numerator: sin²(25°) + sin²(65°) = sin²(25°) + cos²(25°) = 1. Step 4: Rewrite denominator: cos²(25°) + cos²(65°) = cos²(25°) + sin²(25°) = 1. Step 5: Therefore, the expression equals 1/1 = 1.

Question 25

If tan(3α) = cot(2α + 15°), where both angles are acute, find the value of α.

Accepted Answer

Step 1: Using the complementary angle property tan(A) = cot(90° - A), we have cot(2α + 15°) = tan(90° - (2α + 15°)) = tan(75° - 2α). Step 2: Therefore, tan(3α) = tan(75° - 2α). Step 3: Since both angles are acute and the tangent function is one-to-one in the acute range, we have 3α = 75° - 2α. Step 4: Solving for α: 3α + 2α = 75°, so 5α = 75°, which gives α = 15°.

Question 26

If cos(θ) = 3/5 and θ is an acute angle, then what is sin(90° − θ)?

Accepted Answer

Step 1: Recognize that sin(90° − θ) is the sine of the complement of θ. Step 2: Apply the complementary angle identity: sin(90° − θ) = cos(θ). Step 3: Since cos(θ) = 3/5, we have sin(90° − θ) = 3/5.

Question 27

Given that tan(28°) · tan(62°) = p, what is the value of tan(28°) · tan(28°) · tan(62°) · tan(62°)?

Accepted Answer

Step 1: Recognize that 28° and 62° are complementary angles (28° + 62° = 90°). Step 2: Use the complementary angle property: tan(62°) = cot(28°) = 1/tan(28°). Step 3: Therefore, tan(28°) · tan(62°) = tan(28°) · [1/tan(28°)] = 1, so p = 1. Step 4: Calculate tan²(28°) · tan²(62°) = [tan(28°) · tan(62°)]² = p² = 1² = 1.

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