Trigonometric Identities and Inverse Functions

Flashcards covering reciprocal, quotient, Pythagorean, double angle, and sum/difference identities, plus inverse trigonometric compositions and verification steps.

Reciprocal Identity for $\sin(u)$

$$\sin(u) = \frac{1}{\csc(u)}$$

Reciprocal Identity for $\cos(u)$

$$\cos(u) = \frac{1}{\sec(u)}$$

Reciprocal Identity for $\tan(u)$

$$\tan(u) = \frac{1}{\cot(u)}$$

Quotient Identity for $\tan(u)$

$$\tan(u) = \frac{\sin(u)}{\cos(u)}$$

Quotient Identity for $\cot(u)$

$$\cot(u) = \frac{\cos(u)}{\sin(u)}$$

Pythagorean Identity (Sine and Cosine)

$$\sin^2(u) + \cos^2(u) = 1$$

Pythagorean Identity (Tangent and Secant)

$$\tan^2(u) + 1 = \sec^2(u)$$

Pythagorean Identity (Cotangent and Cosecant)

$$\cot^2(u) + 1 = \csc^2(u)$$

Even Function Identities

$\cos(-u) = \cos(u)$ and $\sec(-u) = \sec(u)$ (these functions do not change their sign)

Odd Function Identities

$\sin(-u) = -\sin(u)$, $\tan(-u) = -\tan(u)$, $\csc(-u) = -\csc(u)$, and $\cot(-u) = -\cot(u)$

Steps to Verify Trig Identities

1) Find what is in common, 2) Factor, 3) Rewrite Equation, 4) See how similar, 5) Replace, 6) Convert values to values of $\sin(u)$ and $\cos(u)$, 7) Algebra.

Inverse Sine Range

$\sin^{-1}(x)$ is defined on the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$

Inverse Cosine Range

$\cos^{-1}(x)$ is defined on the interval $[0, \pi]$

Inverse Tangent Range

$\tan^{-1}(x)$ is defined on the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$

Composition: $\cos(\sin^{-1}(x))$

$$\sqrt{1 - x^2}$$

Composition: $\tan(\sin^{-1}(x))$

$$\frac{x}{\sqrt{1 - x^2}}$$

Composition: $\tan(\cos^{-1}(x))$

$$\frac{\sqrt{1 - x^2}}{x}$$

Double Angle Identity for $\sin(2x)$

$$\sin(2x) = 2\sin(x)\cos(x)$$

Double Angle Identities for $\cos(2x)$

$$\cos(2x) = \cos^2(x) - \sin^2(x) = 1 - 2\sin^2(x) = 2\cos^2(x) - 1$$

Double Angle Identity for $\tan(2x)$

$$\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}$$

Sum Identity for $\sin(A + B)$

$$\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$$

Difference Identity for $\sin(A - B)$

$$\sin(A - B) = \sin(A)\cos(B) - \cos(A)\sin(B)$$

Sum Identity for $\cos(A + B)$

$$\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B)$$

Difference Identity for $\cos(A - B)$

$$\cos(A - B) = \cos(A)\cos(B) + \sin(A)\sin(B)$$

Sum Identity for $\tan(A + B)$

$$\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A)\tan(B)}$$

Difference Identity for $\tan(A - B)$

$$\tan(A - B) = \frac{\tan(A) - \tan(B)}{1 + \tan(A)\tan(B)}$$