Trigonometric Expressions

Basic Identities

• sin²x + cos²x = 1
• tg x = sin x / cos x, ctg x = cos x / sin x, tg x · ctg x = 1
• 1 + tg²x = 1 / cos²x, 1 + ctg²x = 1 / sin²x

Addition and Double-Angle Formulas

• sin 2x = 2 sin x cos x
• cos 2x = cos²x − sin²x = 1 − 2sin²x = 2cos²x − 1
• sin(a ± b) = sin a cos b ± cos a sin b
• cos(a ± b) = cos a cos b ∓ sin a sin b
• tg(a ± b) = (tg a ± tg b) / (1 ∓ tg a · tg b)
• ctg(a − b) = (ctg a · ctg b + 1) / (ctg b − ctg a)

Reduction Formulas

sin(180° − x) = sin x, cos(180° − x) = −cos x, tg(180° − x) = −tg x, ctg(180° − x) = −ctg x, sin(90° − x) = cos x. First determine the quadrant, then the sign.

Divide the numerator and denominator by cos x: a homogeneous ratio reduces to tg x. For example, if tg x = 2, then (3sin x + cos x) / (sin x − cos x) = (3 · 2 + 1) / (2 − 1) = 7.

Tables

angle	30°	45°	60°
sin	1/2	√2/2	√3/2
cos	√3/2	√2/2	1/2
tg	√3/3	1	√3

Signs by quadrant (on the unit circle):

Quadrant	sin	cos	tg, ctg
I	+	+	+
II	+	−	−
III	−	−	+
IV	−	+	−

Examples

• sin x = 3/5, x in the first quadrant: cos²x = 1 − 9/25 = 16/25, cos x = 4/5, tg x = 3/4, sin 2x = 24/25, cos 2x = 1 − 18/25 = 7/25. Check: (24/25)² + (7/25)² = 576/625 + 49/625 = 1.
• sin⁴x − cos⁴x = (sin²x − cos²x)(sin²x + cos²x) = sin²x − cos²x, therefore sin⁴x − cos⁴x + 2cos²x = sin²x + cos²x = 1.
• sin 150° + cos 120° + tg 135°: sin 150° = sin 30° = 1/2, cos 120° = −cos 60° = −1/2, tg 135° = −tg 45° = −1. The sum is −1.

Watch Out

• sin(a + b) ≠ sin a + sin b.
• When taking a square root, the quadrant tells you which of ± to choose.
• Sign in reduction: cos 120° < 0.
• sin²x means (sin x)², not sin(x²).
• Of the three forms of cos 2x, choose the convenient one: if sin x is given, use 1 − 2sin²x; if cos x is given, use 2cos²x − 1.