Trigonometry and Polar Coordinates - AP Precalculus

Arc Length = $r \theta$. Where $\theta$ is in radians.

Tangent = Opposite side / Adjacent side. Basic trigonometric ratio in right triangles.

Secant = $\frac{1}{\cos \theta}$. Reciprocal function of cosine.

Sector Area = $\frac{1}{2} r^2 \theta$. Where $\theta$ is in radians.

$(0, 0)$. The pole has radius $0$ and undefined angle.

Cosine = Adjacent side / Hypotenuse. Basic trigonometric ratio in right triangles.

Tangent = Opposite side / Adjacent side. Basic trigonometric ratio in right triangles.

$\sin^2 \theta + \cos^2 \theta = 1$. Fundamental trigonometric identity from unit circle.

Secant = $\frac{1}{\cos \theta}$. Reciprocal function of cosine.

Cosecant = $\frac{1}{\sin \theta}$. Reciprocal function of sine.

Cotangent = $\frac{1}{\tan \theta}$. Reciprocal function of tangent.

$\sin(a + b) = \sin a \cos b + \cos a \sin b$. Used to find sine of sum of two angles.

Sine = Opposite side / Hypotenuse. Basic trigonometric ratio in right triangles.

$\cos(a + b) = \cos a \cos b - \sin a \sin b$. Used to find cosine of sum of two angles.

$\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}$. Used to find tangent of sum of two angles.

$45^\circ$. Use $\frac{\pi}{4} \times \frac{180^\circ}{\pi}$.

$\pi$ radians. Use $180^\circ \times \frac{\pi}{180^\circ}$.

$30^\circ$. Subtract $180^\circ$ from third quadrant angle.

$\frac{1}{2}$. Standard unit circle value.

$\frac{1}{2}$. Standard unit circle value.

1. Standard unit circle value.

Arc Length = $r \theta$. Where $\theta$ is in radians.

$c^2 = a^2 + b^2 - 2ab\cos C$. Relates all three sides and one angle.

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Ratios of sides to opposite angles are equal.

$\tan(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$. Sign depends on quadrant of $\frac{\theta}{2}$.

$\cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$. Sign depends on quadrant of $\frac{\theta}{2}$.

$\sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}}$. Sign depends on quadrant of $\frac{\theta}{2}$.

$\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2 \theta}$. Derived from angle addition formulas.

$\cos(2\theta) = \cos^2 \theta - \sin^2 \theta$. Derived from angle addition formulas.