Addition & Subtraction Formulas
- \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
- \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)
- \( \cos(A + B) = \cos A \cos B - \sin A \sin B \)
- \( \cos(A - B) = \cos A \cos B + \sin A \sin B \)
- \( \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \)
- \( \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \)
Exact Values Using Identities
- Break angle into sum/difference of special angles (\(30°\), \(45°\), \(60°\), \(90°\)).
- Apply the correct identity and substitute known values.
- Use ASTC rule to confirm the sign.
Example: \( \sin 15^\circ = \sin(45^\circ - 30^\circ) = \sin45^\circ\cos30^\circ -
\cos45^\circ\sin30^\circ \)
Special Angles (tan285°, cot285°)
- Reduce using reference angle and quadrant.
- \( \tan285^\circ = -\tan75^\circ \), \( \cot285^\circ = -\cot75^\circ \)
Expansion & Simplification
- Use identities to condense into a single trig function.
- \( \cos80^\circ \cos50^\circ + \sin80^\circ \sin50^\circ = \cos(80^\circ - 50^\circ) = \cos30^\circ \)
Radians
- All formulas work in radians as well as degrees.
- Conversion: degrees = radians \(× 180/π\).
- Example: \(15° = π/12\), \(75° = 5π/12\).
Double-Angle Identities
- \( \sin(2a) = 2\sin a \cos a \)
- \( \cos(2a) = \cos^2 a - \sin^2 a = 2\cos^2 a - 1 = 1 - 2\sin^2 a \)
- \( \tan(2a) = \frac{2\tan a}{1 - \tan^2 a} \)
Using Identities in Problems
- Apply double-angle or half-angle formulas as needed.
- Example: If \( \cos48^\circ = M \), then \( \cos96^\circ = 2M^2 - 1 \).
Proof Strategy
- Work with the more complex side.
- Change to sine and cosine when stuck.
- Use common denominators when adding/subtracting fractions.
- Multiply by conjugates if useful.
- Simplify step by step until both sides match.
Core Proof Tips
- Start with the tougher side.
- Use Pythagorean identity: \( \sin^2x + \cos^2x = 1 \).
- Replace tan, cot, sec, csc with sin/cos.
- Always check for restrictions (denominator \( \neq 0 \)).