Addition & Subtraction Formulas
- \( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \)
- \( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \)
- \( \tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B} \)
Example: \( \sin(75^\circ) = \sin(45^\circ + 30^\circ) = \sin45^\circ\cos30^\circ +
\cos45^\circ\sin30^\circ \)
Angle Addition & Subtraction Identities
- Break the angle into a sum/difference of special angles (\(30°\), \(45°\), \(60°\), \(90°\)).
- Write the expression using a trig identity.
- Substitute known trig values and simplify.
- Check the sign using the ASTC rule.
Example: \( \sin(15^\circ) = \sin(45^\circ - 30^\circ) = \sin45^\circ\cos30^\circ -
\cos45^\circ\sin30^\circ \)
Step-by-Step: Finding Exact Values
- Choose a combination of known angles.
- Write the trig expression in the correct form.
- Apply the correct identity (sin, cos, or tan).
- Simplify using known values.
- Use ASTC to confirm the sign.
Example: \( \tan(75^\circ) = \frac{\tan45^\circ + \tan30^\circ}{1 -
\tan45^\circ\tan30^\circ} = \frac{1 + \frac{1}{\sqrt{3}}}{1 - 1 \cdot \frac{1}{\sqrt{3}}} \)
Reference Angles & Periodicity
- Reduce angles using reference angles and periodicity.
- Example: 285° = 360° - 75°, so it lies in Quadrant IV.
- \( \tan285^\circ = -\tan75^\circ \), \( \cot285^\circ = -\cot75^\circ \)
Expanding & Simplifying Expressions
- Condense trig expressions using formulas.
- \( \cos80^\circ\cos50^\circ + \sin80^\circ\sin50^\circ = \cos(80^\circ - 50^\circ) = \cos30^\circ \)
Radians & Degree Conversion
- Addition and subtraction formulas work in radians too.
- Convert: degrees = radians \(× 180/π\).
- Example: \(15° = π/12\), \(75° = 5π/12\).
Double-Angle Identities
- Derived from addition formulas.
- \( \sin(2a) = 2\sin a\cos a \)
- \( \cos(2a) = \cos^2 a - \sin^2 a = 2\cos^2 a - 1 = 1 - 2\sin^2 a \)
Special Problems & Connections
- Sometimes trig values are connected using identities.
- Example: If \( \cos48^\circ = M \), then \( \cos96^\circ = 2M^2 - 1 \) (double-angle identity).
Trigonometric Proofs
- Work with the more complicated side.
- Convert all functions to sine and cosine if possible.
- Use algebra skills: factoring, simplification.
- Combine fractions with common denominators.
- Multiply by conjugates if needed.
- Simplify until both sides match.
Proof Techniques Summary
- Start with the tougher-looking side.
- Use basic identities: \( \sin^2\theta + \cos^2\theta = 1 \).
- Replace tan, cot, sec, csc with sin and cos if stuck.
- Keep track of restrictions (denominators not zero).
- Be patient — proofs often take multiple steps!