2.2.12 Trigonometric Formulas

Recently, questions involving trigonometric functions have encompassed some basic trigonometric identities. The most popular ones tested are included here:

The Fundamental Identities

$\sin^2 \theta + \cos^2 \theta = 1$ $1 + \cot^2 \theta = \csc^2 \theta$ $\tan^2 \theta + 1 = \sec^2 \theta$

Sum to Difference Formulas

$\sin(a \pm b) = \sin(a)\cos(b) \pm \sin(b)\cos(a)$ $\cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b)$

Double Angle Formulas

$\sin(2a) = 2\sin(a)\cos(a)$ $\cos(2a) = \cos^2(a) - \sin^2(a)$ $\cos(2a) = 1 - 2\sin^2(a)$ $\cos(2a) = 2\cos^2(a) - 1$

Sine → Cosine

$\sin(90^\circ - \theta) = \cos(\theta)$

Example: $\sin(10^\circ)\cos(20^\circ) + \sin(20^\circ)\cos(10^\circ) = \sin(10^\circ + 20^\circ) = \sin(30^\circ) = 1/2$

Problem Set 2.2.12

cos² 30° + sin² 30°

cos² 30° - sin² 30°

2 sin 15° cos 15°

2 sin 30° sin 30° - 1

1 - sin² 30°

cos 22° = sin θ, 0° < θ < 90°, θ =

[2 sin(π/3) cos(π/3)]²

2 sin 15° cos 15° - 1

3 csc² 45° - 3 cot² 45°

cos² 30° - sin² 30°

sin 105° cos 105°

sin 38° = cos θ, 270° < θ < 360°, θ =

sin 30° cos 60° - sin 60° cos 30°

2 cos²(π/6) - 1

(1 - sin 60°)(1 + sin 60°)

2 sin 15° sin 75°

(sin(π/3) - cos(π/3))(sin(π/3) + cos(π/3))

If sin(A) = 3/5, then cos(2A) =

1 - 2 sin²(π/6)

cos 75° sin 75°

sin 15° cos 45° - sin 45° cos 15°

2 - 4 sin² 30°

cos 95° cos 25° - sin 95° sin 25°

sin(π/6) + cos(π/3)

cos 15° sin 45° - cos 45° sin 15°

(sin(π/6) - cos(π/6))(sin(π/6) + cos(π/6))

2 tan² θ - 2 sec² θ