E: Trigonometry flashcards

Topic 6: E: Trigonometry

sin⁻¹x vs (sin x)⁻¹

arcsin x is the inverse function; cosec x = 1/sin x.

Degrees vs radians

Arc length, sector area, small angles and calculus all need radians.

Identity (≡) vs equation (=)

An identity is true for all values; an equation is solved for particular ones.

Principal value vs all solutions

arcsin, arccos and arctan return one value; the question usually wants all in the interval.

Small-angle approximations (θ in radians), and sin 4θ and cos 3θ

sin θ ≈ θ, cos θ ≈ 1 − θ²/2, tan θ ≈ θ. Substitute the whole argument: sin 4θ ≈ 4θ, cos 3θ ≈ 1 − 9θ²/2.

Arc length and sector area

s = rθ and A = ½r²θ, with θ in radians.

Sine rule, cosine rule and area of a triangle

a/sin A = b/sin B = c/sin C a² = b² + c² − 2bc cos A Area = ½ab sin C

The three Pythagorean identities

cos²A + sin²A ≡ 1 sec²A ≡ 1 + tan²A cosec²A ≡ 1 + cot²A

Double-angle formulae

sin 2A ≡ 2 sin A cos A cos 2A ≡ cos²A − sin²A ≡ 2cos²A − 1 ≡ 1 − 2sin²A tan 2A ≡ 2 tan A/(1 − tan²A)

Compound-angle formulae for sin(A ± B) and cos(A ± B)

sin(A ± B) ≡ sin A cos B ± cos A sin B cos(A ± B) ≡ cos A cos B ∓ sin A sin B

How do you write a cos θ + b sin θ as R cos(θ − α)?

Expand: R cos θ cos α + R sin θ sin α. Match R cos α = a, R sin α = b. R = √(a² + b²), tan α = b/a.

Signs of sin, cos and tan in each quadrant (0 to 2π)

First quadrant: all positive. Second: sin positive. Third: tan positive. Fourth: cos positive.