Trigonometry Review

1. Pythagorean Identities

These come from the Pythagorean Theorem x² + y² = 1 on the unit circle.

sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

These are fundamental and should be memorized — they are used often in simplifying and proving other identities.

2. Reciprocal Identities

Each reciprocal identity defines one trigonometric function as the reciprocal of another.

sin θ = 1 / csc θ, csc θ = 1 / sin θ
cos θ = 1 / sec θ, sec θ = 1 / cos θ
tan θ = 1 / cot θ, cot θ = 1 / tan θ

These are useful when rewriting problems in terms of sine and cosine.

3. Quotient Identities

These come directly from the definitions of tangent and cotangent.

tan θ = sin θ / cos θ
cot θ = cos θ / sin θ

They help connect sine and cosine to the other trig functions.

4. Co-Function Identities

Co-function identities show relationships between complementary angles. They are based on the idea that sin θ = cos(90° − θ).

sin θ = cos(90° − θ)
cos θ = sin(90° − θ)
tan θ = cot(90° − θ)
csc θ = sec(90° − θ)
sec θ = csc(90° − θ)
cot θ = tan(90° − θ)

6. Sum and Difference Formulas

These let you find the trig values of combined angles.

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Use the top signs together (+ with +, – with –), and the bottom signs opposite for cosine and tangent.

7. Double-Angle and Half-Angle Formulas

Double-Angle:
sin(2θ) = 2 sin θ cos θ
cos(2θ) = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ
tan(2θ) = (2 tan θ) / (1 − tan²θ)

Half-Angle:
sin(θ/2) = ±√((1 − cos θ) / 2)
cos(θ/2) = ±√((1 + cos θ) / 2)
tan(θ/2) = (1 − cos θ) / sin θ = sin θ / (1 + cos θ)

(The ± sign depends on the quadrant in which θ/2 lies.)