Introduction to Trigonometry — Practice

Question 1

If sin A = 3/5, find cos A.

Accepted Answer

✦ Answer: sin²A + cos²A = 1; (3/5)² + cos²A = 1; cos²A = 1 − 9/25 = 16/25; cos A = 4/5.

Question 2

What is tan 60°?

Accepted Answer

✦ Answer: tan 60° = √3 (from memorised table).

Question 3

Prove: (1 − cos²θ)(1 + cot²θ) = 1.

Accepted Answer

Step 1 — Use identities.
  (1 − cos²θ) = sin²θ
  (1 + cot²θ) = cosec²θ = 1/sin²θ

Step 2 — Multiply.
  LHS = sin²θ × 1/sin²θ = 1 = RHS ✓

✦ Proved.

Question 4

If 5 sin θ = 3, find the values of cos θ, tan θ, sec θ, cosec θ, cot θ.

Accepted Answer

Step 1 — Find sin θ.
  5 sin θ = 3 ⟹ sin θ = 3/5

Step 2 — Find cos θ using identity.
  sin²θ + cos²θ = 1
  9/25 + cos²θ = 1
  cos²θ = 16/25
  cos θ = 4/5 (positive since θ is acute)

Step 3 — Find tan θ.
  tan θ = sin θ / cos θ = (3/5)/(4/5) = 3/4

Step 4 — Find reciprocal ratios.
  cosec θ = 1/sin θ = 5/3
  sec θ = 1/cos θ = 5/4
  cot θ = 1/tan θ = 4/3

✦ Answer: cos θ = 4/5, tan θ = 3/4, sec θ = 5/4, cosec θ = 5/3, cot θ = 4/3.

Practice — Introduction to Trigonometry