Trig Functions

Sine Function
Cosine Function
Tangent Function

Trig Identities

Fundamental Identities (8 of them)
Reciprocal (3)
Quotient (2)
Pythagorean (3)
Reciprocal Identities
$$ \sec{\theta} = \frac{1}{\cos{\theta}} $$
$$ \cot{\theta} = \frac{1}{\tan{\theta}} $$
$$ \csc{\theta} = \frac{1}{\sin{\theta}} $$
Quotient Identities
$$ \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} $$
$$ \cot{\theta} = \frac{\cos{\theta}}{\sin{\theta}} $$
Pythagorean Identities
$$ \sin^2{\theta} + \cos^2{\theta} = 1 $$
$$ 1 + \tan^2{\theta} = \sec^2{\theta} $$
$$ 1 + \cot^2{\theta} = \csc^2{\theta} $$
Sum and Difference Identities
(if you know the rhyme...) "Sine Cosine Cosine Sine, Cosine Cosine SIGN Sine Sine"
$$ \sin{(\theta \pm \phi)} = \sin{\theta} \cos{\phi} \pm \cos{\theta} \sin{\phi} $$
$$ \cos{(\theta \pm \phi)} = \cos{\theta} \cos{\phi} \pm \sin{\theta} \sin{\phi} $$
$$ \tan{(\theta \pm \phi)} = \frac{\sin{(\theta \pm \phi)}}{\cos{(\theta \pm \phi)}} = \frac{\tan{\theta} \pm \tan{\phi}}{1 \pm \tan{\theta} \tan{\phi}} $$
Double Angle Identities
$$ \sin{2\theta} = 2\sin{\theta} \cos{\theta} $$
$$ \cos{2\theta} = \cos^2{\theta} - \sin^2{\theta} = 2\cos^2{\theta} - 1 = 1 - 2\sin^2{\theta} $$
$$ \tan{2\theta} = \frac{2\tan{\theta}}{1 - \tan^2{\theta}} $$
Negative Angle Identities (Odd)
$$ \sin{(-\theta)} = -\sin{\theta} $$
$$ \tan{(-\theta)} = -\tan{\theta} $$
$$ \cot{(-\theta)} = -\cot{\theta} $$
$$ \csc{(-\theta)} = -\csc{\theta} $$
Negative Angle Identities (Even)
$$ \cos{(-\theta)} = \cos{\theta} $$
$$ \sec{(-\theta)} = \sec{\theta} $$
Half Angle Identities
$$ \sin{(\frac{\theta}{2})} = \pm \sqrt{\frac{1 - \cos{\theta}}{2}} $$
$$ \cos{(\frac{\theta}{2})} = \pm \sqrt{\frac{1 + \cos{\theta}}{2}} $$
$$ \tan{(\frac{\theta}{2})} = \pm \sqrt{\frac{1 - \cos{\theta}}{1 + \cos \theta}} = \frac{1 - \cos{\theta}}{\sin{\theta}} = \frac{\sin{\theta}}{1 + \cos{\theta}} $$
Product to Sum
$$ \sin{\alpha} \sin{\beta} = \frac{\cos{(\alpha - \beta)} - \cos{(\alpha + \beta)}}{2} $$
$$ \cos{\alpha} \cos{\beta} = \frac{\cos{(\alpha - \beta)} + \cos{(\alpha + \beta)}}{2} $$
$$ \sin{\alpha} \cos{\beta} = \frac{\sin{(\alpha - \beta)} + \sin{(\alpha - \beta)}}{2} $$
Sum to Product
$$ \sin{\theta} \pm \sin{\phi} = 2\sin{(\frac{\theta \pm \phi}{2})} \cos{(\frac{\theta \pm \phi}{2})} $$
$$ \cos{\theta} + \cos{\phi} = 2\cos{(\frac{\theta + \phi}{2})} \cos{(\frac{\theta - \phi}{2})} $$
$$ \cos{\theta} - \cos{\phi} = -2\sin{(\frac{\theta + \phi}{2})} \sin{(\frac{\theta - \phi}{2})} $$
Example Problem: $$ \cos{15^\circ} $$
$$ = \cos{(60^\circ - 45^\circ)} $$
$$ = \cos{60^\circ} \cos{45^\circ} + \sin{60^\circ} \sin{45^\circ} $$
$$ = \frac{1}{2} \times \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \times \frac{\sqrt{2}}{2} $$
$$ = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} $$
$$ = \frac{\sqrt{2} + \sqrt{6}}{4} $$
When Asked for Verification
Work only on one side to get it to equal the other
When Asked for Trig Equations
Work on both sides to get them equal
Extra
$$ \sin{\theta} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$

Same vs Co

Same trig function
Q II Q I
$$ \theta = 180^\circ - \rho = \pi - \rho $$ $$ \theta = \rho $$
$$ \theta = 180^\circ + \rho = \pi + \rho $$ $$ \theta = 360^\circ - \rho = \pi - \rho $$
Q III Q IV
CO trig function
Q II Q I
$$ \theta = 90^\circ + \rho = \frac{\pi}{2} + \rho $$ $$ \theta = 90^\circ - \rho = \frac{\pi}{2} - \rho $$
$$ \theta = 270^\circ - \rho = \frac{3\pi}{2} - \rho $$ $$ \theta = 270^\circ + \rho = \frac{3\pi}{2} + \rho $$
Q III Q IV

Domains and Ranges

Sine
Domain: $$ [-\frac{\pi}{2}, \frac{\pi}{2}] $$
Range: $$ [-1, 1] $$
Cosine
Domain: $$ [0, \pi] $$
Range: $$ [-1, 1] $$
Tangent
Domain: $$ (-\frac{\pi}{2}, \frac{\pi}{2}) $$
Range: $$ (-\infty, \infty) $$
Cosecant
Domain: $$ [-\frac{\pi}{2}, \frac{\pi}{2}], \theta \neq 0$$
Range: $$ (-\infty, -1], [1, \infty) $$
Secant
Domain: $$ [0, \pi], \theta \neq 0$$
Range: $$ (-\infty, -1], [1, \infty) $$
Cotangent
Domain: $$ (0, \pi) $$
Range: $$ (-\infty, \infty) $$