AP Precalculus: Trigonometry Formulas & Rules
1. Radian/Degree Conversion, Arc Length
- Radians to degrees: \( \theta^\circ = \theta \cdot \frac{180}{\pi} \)
- Degrees to radians: \( \theta = \theta^\circ \cdot \frac{\pi}{180} \)
- Arc length: \( s = r\theta \) (with \( \theta \) in radians)
- Sector area: \( A = \frac{1}{2} r^2 \theta \)
2. Quadrants, Coterminal & Reference Angles
- Quadrants: I (\(0^\circ\)–\(90^\circ\)), II (\(90^\circ\)–\(180^\circ\)), III (\(180^\circ\)–\(270^\circ\)), IV (\(270^\circ\)–\(360^\circ\))
- Coterminal: \( \theta_{\text{coterminal}} = \theta \pm 360^\circ \) or \( \theta \pm 2\pi \)
- Reference angle: smallest angle to x-axis;
- Quadrant II: \( 180^\circ-\theta \) or \( \pi-\theta \)
- Quadrant III: \( \theta-180^\circ \) or \( \theta-\pi \)
- Quadrant IV: \( 360^\circ-\theta \) or \( 2\pi-\theta \)
3. Trig Ratios in Right Triangles
- \( \sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} \)
- \( \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- \( \tan\theta = \frac{\text{opposite}}{\text{adjacent}} \)
- \( \csc\theta = \frac{1}{\sin\theta} \), \( \sec\theta = \frac{1}{\cos\theta} \), \( \cot\theta = \frac{1}{\tan\theta} \)
4. Trig Ratios on the Unit Circle
- Any angle \( \theta \), point \( (x, y) \) on unit circle:
- \( x = \cos\theta \), \( y = \sin\theta \)
- \( \tan\theta = \frac{y}{x} \) if \( x \ne 0 \)
- Pythagorean ID: \( \sin^2\theta+\cos^2\theta=1 \)
5. Special Angles and Reference Angles
- \( 30^\circ, 45^\circ, 60^\circ \) (e.g. \( \sin 45^\circ = \frac{\sqrt{2}}{2} \), \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) )
- Use reference angle plus quadrant sign
- All trig ratios can be determined for special angles via unit circle/triangle
6. Inverse Trigonometric Functions
- \( y = \sin^{-1}(x) \Longleftrightarrow x = \sin y \), \( -1 \leq x \leq 1 \), \( -\frac{\pi}{2} \leq y \leq \frac{\pi}{2} \)
- \( y = \cos^{-1}(x) \Longleftrightarrow x = \cos y \), \( 0 \leq y \leq \pi \)
- \( y = \tan^{-1}(x) \Longleftrightarrow x = \tan y \), \( -\frac{\pi}{2} < y < \frac{\pi}{2} \)
7. Solve Right Triangles & Trig Equations
- Given a side & angle, use trig ratios to solve for missing values
- \( a = c\sin A \), \( b = c\cos A \), \( \tan A = \frac{a}{b} \)
- Solve equations: Isolate function, inverse to get angle, check all valid solutions in period (radians/degrees, as needed)
8. Law of Sines, Law of Cosines, Triangle Area
- Law of Sines: \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
- Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos C \)
- Area (sine): \( A = \frac{1}{2} ab\sin C \)
- Heron's formula: \( s = \frac{a+b+c}{2} \), \( A = \sqrt{s(s-a)(s-b)(s-c)} \)