MAT pure2&3 trigonometry incompl

Question 1

What are the reciprocal trigonometric functions?

Answer 1

1. cosec(θ) = 1 / sin(θ), where sin(θ) ≠ 0.
2. sec(θ) = 1 / cos(θ), where cos(θ) ≠ 0.
3. cot(θ) = 1 / tan(θ), where tan(θ) ≠ 0

Question 2

How can you remember the reciprocal trigonometric functions?

Answer 2

Use the third letter of each function:
- “cosec” has “s” → cosec(θ) = 1 / sin(θ).
- “sec” has “c” → sec(θ) = 1 / cos(θ).
- “cot” has “t” → cot(θ) = 1 / tan(θ).

Question 3

Example: What is cot(120°)?

Answer 3

cot(120°) = 1 / tan(120°)
tan(120°) = -tan(60°) = -√3
cot(120°) = -1/√3

Question 4

What are the Pythagorean identities in trigonometry?

Answer 4

1. sin²(θ) + cos²(θ) ≡ 1.
2. 1 + tan²(θ) ≡ sec²(θ).
3. 1 + cot²(θ) ≡ cosec²(θ).

Question 5

How are the Pythagorean identities derived?

Answer 5

Divide sin²(θ) + cos²(θ) = 1 by:
1. cos²(θ) to get 1 + tan²(θ) = sec²(θ).
2. sin²(θ) to get 1 + cot²(θ) = cosec²(θ).

Question 6

Example: In a triangle, A = 90°, cosec(B) = 2. Find angles B and C.

Answer 6

cosec(B) = 2 → sin(B) = 1/2 → B = 30°.
C = 180° - 90° - 30° = 60°.

Question 7

Verify the identity tan²(C) + 1 ≡ sec²(C) for C = 60°.

Answer 7

tan²(60°) = 3 → tan²(C) + 1 = 3 + 1 = 4.
sec²(60°) = 4.
Hence, tan²(C) + 1 ≡ sec²(C).

Question 8

Example: Solve 2cosec²(θ) = 3 + cot²(θ) for -π ≤ θ ≤ π.

Answer 8

1. Use the identity cosec²(θ) = 1 + cot²(θ).
   2(1 + cot²(θ)) = 3 + cot²(θ).
2. Solve: cot²(θ) - 2cot(θ) - 1 = 0.
   (cot(θ) - 2)(cot(θ) + 1) = 0.
3. Solutions: cot(θ) = 2, cot(θ) = -1.
   θ = arccot(2), θ = arccot(-1).

Question 9

What are the roots of the equation 2cosec²(θ) = 3 + cot²(θ) in radians?

Answer 9

θ = 0.464, θ = π/4, θ = -2.68, θ = -3π/4.

Question 10

What is the period of tan(θ) and cot(θ)?

Answer 10

Both tan(θ) and cot(θ) have a period of π radians.