Introduction to Trigonometry — Practice

Question 1

Write the value of sin 30° + cos 60°.

Accepted Answer

Step 1 — sin 30° = 1/2, cos 60° = 1/2.

Step 2 — Sum = 1/2 + 1/2 = 1.

✦ Answer: 1.

Question 2

Evaluate sin 35° / cos 55°.

Accepted Answer

Step 1 — cos 55° = cos(90° − 35°) = sin 35°.

Step 2 — So sin 35° / cos 55° = sin 35° / sin 35° = 1.

✦ Answer: 1.

Question 3

If sin θ = 3/5, find cosec θ.

Accepted Answer

Step 1 — cosec θ = 1/sin θ.

Step 2 — = 1 ÷ (3/5) = 5/3.

✦ Answer: cosec θ = 5/3.

Question 4

If tan θ = 4/3, find sin θ and cos θ.

Accepted Answer

Step 1 — tan θ = opp/adj = 4/3 → opp = 4k, adj = 3k.

Step 2 — hyp = √((4k)² + (3k)²) = √(25k²) = 5k.

Step 3 — sin θ = 4k/5k = 4/5; cos θ = 3k/5k = 3/5.

✦ Answer: sin θ = 4/5, cos θ = 3/5.

Question 5

Evaluate 2 tan²45° + cos²30° − sin²60°.

Accepted Answer

Step 1 — tan 45° = 1, cos 30° = √3/2, sin 60° = √3/2.

Step 2 — 2(1)² + (√3/2)² − (√3/2)² = 2 + 3/4 − 3/4.

Step 3 — = 2.

✦ Answer: 2.

Question 6

If sec θ = 13/12, find tan θ using an identity.

Accepted Answer

Step 1 — Use 1 + tan²θ = sec²θ.

Step 2 — tan²θ = sec²θ − 1 = (13/12)² − 1 = 169/144 − 144/144 = 25/144.

Step 3 — tan θ = 5/12 (taking the positive value for an acute angle).

✦ Answer: tan θ = 5/12.

Question 7

Prove that (1 − sin θ)/(1 + sin θ) = (sec θ − tan θ)².

Accepted Answer

Step 1 — Start from the RHS: (sec θ − tan θ)² = (1/cos θ − sin θ/cos θ)² = ((1 − sin θ)/cos θ)².

Step 2 — = (1 − sin θ)² / cos²θ. Replace cos²θ = 1 − sin²θ = (1 − sin θ)(1 + sin θ).

Step 3 — = (1 − sin θ)² / [(1 − sin θ)(1 + sin θ)] = (1 − sin θ)/(1 + sin θ) = LHS.

✦ Answer: RHS simplifies to the LHS, hence proved. ∎

Question 8

Prove that (sin θ + cosec θ)² + (cos θ + sec θ)² = 7 + tan²θ + cot²θ.

Accepted Answer

Step 1 — Expand: sin²θ + 2 sinθ cosecθ + cosec²θ + cos²θ + 2 cosθ secθ + sec²θ.

Step 2 — sinθ·cosecθ = 1 and cosθ·secθ = 1, so the cross terms give 2 + 2 = 4.

Step 3 — (sin²θ + cos²θ) = 1; cosec²θ = 1 + cot²θ; sec²θ = 1 + tan²θ.

Step 4 — Total = 1 + 4 + (1 + cot²θ) + (1 + tan²θ) = 7 + tan²θ + cot²θ = RHS.

✦ Answer: LHS = 7 + tan²θ + cot²θ, hence proved. ∎

Question 9

Without using tables, evaluate (cos 38° cos 52° − sin 38° sin 52°).

Accepted Answer

Step 1 — cos 52° = cos(90° − 38°) = sin 38°; sin 52° = sin(90° − 38°) = cos 38°.

Step 2 — Substitute: cos 38°·sin 38° − sin 38°·cos 38°.

Step 3 — = 0.

✦ Answer: 0.

Question 10

If 3 cot θ = 4, find the value of (5 sin θ − 3 cos θ)/(5 sin θ + 3 cos θ).

Accepted Answer

Step 1 — cot θ = 4/3 ⇒ tan θ = 3/4. So opp = 3k, adj = 4k, hyp = 5k.

Step 2 — sin θ = 3/5, cos θ = 4/5.

Step 3 — Numerator: 5(3/5) − 3(4/5) = 3 − 12/5 = 3/5.

Step 4 — Denominator: 5(3/5) + 3(4/5) = 3 + 12/5 = 27/5.

Step 5 — Ratio = (3/5)/(27/5) = 3/27 = 1/9.

✦ Answer: 1/9.

Practice — Introduction to Trigonometry

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