Trigonometry Formulas Vocabulary

Flashcards based on the trigonometry formulas provided in the lecture notes, covering odd-even, sum and difference, cofunction, double-angle, and half-angle identities, plus law of sines/cosines and triangle area formulas.

Odd-Even Identity for $\sin(-x)$

$$- \sin(x)$$

Odd-Even Identity for $\csc(-x)$

$$- \csc(x)$$

Odd-Even Identity for $\cos(-x)$

$$\cos(x)$$

Odd-Even Identity for $\sec(-x)$

$$\sec(x)$$

Odd-Even Identity for $\tan(-x)$

$$- \tan(x)$$

Odd-Even Identity for $\cot(-x)$

$$- \cot(x)$$

Sum Identity: $\sin(u+v)$

$$\sin(u)\cos(v) + \cos(u)\sin(v)$$

Difference Identity: $\sin(u-v)$

$$\sin(u)\cos(v) - \cos(u)\sin(v)$$

Sum Identity: $\cos(u+v)$

$$\cos(u)\cos(v) - \sin(u)\sin(v)$$

Difference Identity: $\cos(u-v)$

$$\cos(u)\cos(v) + \sin(u)\sin(v)$$

Sum Identity: $\tan(u+v)$

$$\frac{\tan(u) + \tan(v)}{1 - \tan(u)\tan(v)}$$

Difference Identity: $\tan(u-v)$

$$\frac{\tan(u) - \tan(v)}{1 + \tan(u)\tan(v)}$$

Cofunction Identity for $\cos\left(\frac{\pi}{2} - u\right)$

$$\sin(u)$$

Cofunction Identity for $\sin\left(\frac{\pi}{2} - u\right)$

$$\cos(u)$$

Cofunction Identity for $\tan\left(\frac{\pi}{2} - u\right)$

$$\cot(u)$$

Cofunction Identity for $\cot\left(\frac{\pi}{2} - u\right)$

$$\tan(u)$$

Cofunction Identity for $\sec\left(\frac{\pi}{2} - u\right)$

$$\csc(u)$$

Cofunction Identity for $\csc\left(\frac{\pi}{2} - u\right)$

$$\sec(u)$$

Double-Angle Identity for $\sin(2u)$

$$2\sin(u)\cos(u)$$

Double-Angle Identities for $\cos(2u)$

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

Double-Angle Identity for $\tan(2u)$

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

Power-Reducing Identity for $\sin^2(u)$

$$\frac{1 - \cos(2u)}{2}$$

Power-Reducing Identity for $\cos^2(u)$

$$\frac{1 + \cos(2u)}{2}$$

Power-Reducing Identity for $\tan^2(u)$

$$\frac{1 - \cos(2u)}{1 + \cos(2u)}$$

Half-Angle Identity for $\sin\left(\frac{u}{2}\right)$

$$\pm \sqrt{\frac{1 - \cos(u)}{2}}$$

Half-Angle Identity for $\cos\left(\frac{u}{2}\right)$

$$\pm \sqrt{\frac{1 + \cos(u)}{2}}$$

Half-Angle Identities for $\tan\left(\frac{u}{2}\right)$

$$\pm \sqrt{\frac{1 - \cos(u)}{1 + \cos(u)}} = \frac{1 - \cos(u)}{\sin(u)} = \frac{\sin(u)}{1 + \cos(u)}$$

Law of Sines

$$\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}$$

Law of Cosines (for side a)

$$a^2 = b^2 + c^2 - 2bc \cos(A)$$

Triangle Area (Trigonometric)

$$\text{Area} = \frac{1}{2}bc\sin(A) = \frac{1}{2}ac\sin(B) = \frac{1}{2}ab\sin(C)$$

Heron's Formula

$$\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$$

Semi-perimeter ($s$)

$$s = \frac{1}{2}(a + b + c)$$

Trigonometric Form of a Complex Number

$$z = a + bi = r(\cos(\theta) + i\sin(\theta))$$