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Right triangle definition
For this definition we assume that

$$ 0 < \theta < \frac{\pi}{2} \quad \text{or} \quad 0^\circ < \theta < 90^\circ $$

Unit circle definition
For this definition 胃 is any angle.

$$ \sin \theta = \frac{y}{1} = y \qquad \csc \theta = \frac{1}{y} $$ $$ \cos \theta = \frac{x}{1} = x \qquad \sec \theta = \frac{1}{x} $$ $$ \tan \theta = \frac{y}{x} \qquad \cot \theta = \frac{x}{y} $$

Domain
The domain is all the values of 胃 that
can be plugged into the function.

$$ \sin \theta,\quad \theta \text{ can be any angle} $$ $$ \cos \theta,\quad \theta \text{ can be any angle} $$ $$ \tan \theta,\quad \theta \ne \left(n + \frac{1}{2}\right)\pi,\; n = 0, \pm 1, \pm 2, \ldots $$ $$ \csc \theta,\quad \theta \ne n\pi,\; n = 0, \pm 1, \pm 2, \ldots $$ $$ \sec \theta,\quad \theta \ne \left(n + \frac{1}{2}\right)\pi,\; n = 0, \pm 1, \pm 2, \ldots $$ $$ \cot \theta,\quad \theta \ne n\pi,\; n = 0, \pm 1, \pm 2, \ldots $$

Range
The range is all possible values to get
out of the function.

$$ -1 \le \sin \theta \le 1 \qquad \csc \theta \ge 1 \;\text{and}\; \csc \theta \le -1 $$ $$ -1 \le \cos \theta \le 1 \qquad \sec \theta \ge 1 \;\text{and}\; \sec \theta \le -1 $$ $$ -\infty < \tan \theta < \infty \qquad -\infty < \cot \theta < \infty $$

Period
The period of a function is the number, T, such that f(胃 + T) = f(胃). So, if 蝇 is a fixed number and 胃 is any angle, we have the following periods.

$$ \sin(\omega \theta) \;\rightarrow\; T = \frac{2\pi}{\omega} $$ $$ \cos(\omega \theta) \;\rightarrow\; T = \frac{2\pi}{\omega} $$ $$ \tan(\omega \theta) \;\rightarrow\; T = \frac{\pi}{\omega} $$ $$ \csc(\omega \theta) \;\rightarrow\; T = \frac{2\pi}{\omega} $$ $$ \sec(\omega \theta) \;\rightarrow\; T = \frac{2\pi}{\omega} $$ $$ \cot(\omega \theta) \;\rightarrow\; T = \frac{\pi}{\omega} $$

Tangent and Cotangent Identities

$$ \tan \theta = \frac{\sin \theta}{\cos \theta} \qquad \cot \theta = \frac{\cos \theta}{\sin \theta} $$

Reciprocal Identities

$$ \sec \theta = \frac{1}{\cos \theta} \qquad \cos \theta = \frac{1}{\sec \theta} $$ $$ \cot \theta = \frac{1}{\tan \theta} \qquad \tan \theta = \frac{1}{\cot \theta} $$

Pythagorean Identities

$$ \sin^2 \theta + \cos^2 \theta = 1 $$ $$ \tan^2 \theta + 1 = \sec^2 \theta $$ $$ 1 + \cot^2 \theta = \csc^2 \theta $$

Even/Odd Formulas

$$ \sin(-\theta) = -\sin \theta \qquad \csc(-\theta) = -\csc \theta $$ $$ \cos(-\theta) = \cos \theta \qquad \sec(-\theta) = \sec \theta $$ $$ \tan(-\theta) = -\tan \theta \qquad \cot(-\theta) = -\cot \theta $$

Periodic Formulas
If n is an integer.

$$ \sin(\theta + 2\pi n) = \sin \theta \qquad \csc(\theta + 2\pi n) = \csc \theta $$ $$ \cos(\theta + 2\pi n) = \cos \theta \qquad \sec(\theta + 2\pi n) = \sec \theta $$ $$ \tan(\theta + \pi n) = \tan \theta \qquad \cot(\theta + \pi n) = \cot \theta $$

Double Angle Formulas

$$ \sin(2\theta) = 2 \sin \theta \cos \theta $$ $$ \cos(2\theta) = \cos^2 \theta – \sin^2 \theta $$ $$ \cos(2\theta) = 2 \cos^2 \theta – 1 $$ $$ \cos(2\theta) = 1 – 2 \sin^2 \theta $$ $$ \tan(2\theta) = \frac{2 \tan \theta}{1 – \tan^2 \theta} $$

Degrees to Radians Formulas
If x is an angle in degrees and t is an
angle in radians then

$$ \frac{\pi}{180} = \frac{t}{x} \;\Rightarrow\; t = \frac{\pi x}{180} \;\text{and}\; x = \frac{180t}{\pi} $$

Sum and Difference Formulas

$$ \sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta $$ $$ \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta $$ $$ \tan(\alpha \pm \beta) = \frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta} $$

Product to Sum Formulas

$$ \sin \alpha \sin \beta = \frac{1}{2}\bigl[\cos(\alpha – \beta) – \cos(\alpha + \beta)\bigr] $$ $$ \cos \alpha \cos \beta = \frac{1}{2}\bigl[\cos(\alpha – \beta) + \cos(\alpha + \beta)\bigr] $$ $$ \sin \alpha \cos \beta = \frac{1}{2}\bigl[\sin(\alpha + \beta) + \sin(\alpha – \beta)\bigr] $$ $$ \cos \alpha \sin \beta = \frac{1}{2}\bigl[\sin(\alpha + \beta) – \sin(\alpha – \beta)\bigr] $$

Sum to Product Formulas

$$ \sin \alpha + \sin \beta = 2 \sin\!\left(\frac{\alpha + \beta}{2}\right) \cos\!\left(\frac{\alpha – \beta}{2}\right) $$ $$ \sin \alpha – \sin \beta = 2 \cos\!\left(\frac{\alpha + \beta}{2}\right) \sin\!\left(\frac{\alpha – \beta}{2}\right) $$ $$ \cos \alpha + \cos \beta = 2 \cos\!\left(\frac{\alpha + \beta}{2}\right) \cos\!\left(\frac{\alpha – \beta}{2}\right) $$ $$ \cos \alpha – \cos \beta = -2 \sin\!\left(\frac{\alpha + \beta}{2}\right) \sin\!\left(\frac{\alpha – \beta}{2}\right) $$

Cofunction Formulas

$$ \sin\!\left(\frac{\pi}{2} – \theta\right) = \cos \theta \qquad \cos\!\left(\frac{\pi}{2} – \theta\right) = \sin \theta $$ $$ \csc\!\left(\frac{\pi}{2} – \theta\right) = \sec \theta \qquad \sec\!\left(\frac{\pi}{2} – \theta\right) = \csc \theta $$ $$ \tan\!\left(\frac{\pi}{2} – \theta\right) = \cot \theta \qquad \cot\!\left(\frac{\pi}{2} – \theta\right) = \tan \theta $$

Definition
y = sin^鈭�1 x is equivalent to x = sin y
y = cos^鈭�1 x is equivalent to x = cos y
y = tan^鈭�1 x is equivalent to x = tan y

Domain and Range

$$ \begin{array}{c c c} \textbf{Function} & \textbf{Domain} & \textbf{Range} \\[6pt] y = \sin^{-1} x & -1 \le x \le 1 & -\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2} \\[6pt] y = \cos^{-1} x & -1 \le x \le 1 & 0 \le y \le \pi \\[6pt] y = \tan^{-1} x & -\infty < x < \infty & -\dfrac{\pi}{2} < y < \dfrac{\pi}{2} \end{array} $$

Inverse Properties

$$ \cos\!\bigl(\cos^{-1}(x)\bigr) = x \qquad \cos^{-1}\!\bigl(\cos(\theta)\bigr) = \theta $$ $$ \sin\!\bigl(\sin^{-1}(x)\bigr) = x \qquad \sin^{-1}\!\bigl(\sin(\theta)\bigr) = \theta $$ $$ \tan\!\bigl(\tan^{-1}(x)\bigr) = x \qquad \tan^{-1}\!\bigl(\tan(\theta)\bigr) = \theta $$

Alternate Notation

$$ \sin^{-1} x = \arcsin x $$ $$ \cos^{-1} x = \arccos x $$ $$ \tan^{-1} x = \arctan x $$

Law of Sines

$$ \frac{\sin \alpha}{a} = \frac{\sin \beta}{b} = \frac{\sin \gamma}{c} $$

Law of Cosines

$$ a^2 = b^2 + c^2 – 2bc \cos \alpha $$ $$ b^2 = a^2 + c^2 – 2ac \cos \beta $$ $$ c^2 = a^2 + b^2 – 2ab \cos \gamma $$

Mollweide鈥檚 Formula

$$ \frac{a + b}{c} = \frac{\cos\!\left(\tfrac{1}{2}(\alpha – \beta)\right)} {\sin\!\left(\tfrac{1}{2}\gamma\right)} $$

Law of Tangents

$$ \frac{a – b}{a + b} = \frac{\tan\!\left(\tfrac{1}{2}(\alpha – \beta)\right)} {\tan\!\left(\tfrac{1}{2}(\alpha + \beta)\right)} $$ $$ \frac{b – c}{b + c} = \frac{\tan\!\left(\tfrac{1}{2}(\beta – \gamma)\right)} {\tan\!\left(\tfrac{1}{2}(\beta + \gamma)\right)} $$ $$ \frac{a – c}{a + c} = \frac{\tan\!\left(\tfrac{1}{2}(\alpha – \gamma)\right)} {\tan\!\left(\tfrac{1}{2}(\alpha + \gamma)\right)} $$