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tags: 數學
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# 三角函數
- Fundamental Trigonometric Identities
- Power Reducing Formulas

## Fucdamental Trigonometric Identities
### Reciprocal identities (倒數)

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

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

### Quotient identities (商)

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

### Pythagorean identities (平方關係)

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

### Cofunction identities (餘數)

$sin(\frac{\pi}{2}-\theta)=cos\theta$
$cos(\frac{\pi}{2}-\theta)=sin\theta$

$tan(\frac{\pi}{2}-\theta)=cot\theta$
$cot(\frac{\pi}{2}-\theta)=tan\theta$

$sec(\frac{\pi}{2}-\theta)=csc\theta$
$csc(\frac{\pi}{2}-\theta)=sec\theta$

### Odd/Even function identities (奇/偶函數)
$sin(-\theta)=-sin\theta$
$cos(-\theta)=cos\theta$
$tan(-\theta)=-tan\theta$

$csc(-\theta)=-csc\theta$
$sec(-\theta)=sec\theta$
$cot(-\theta)=-cot\theta$

## Power Reducing Formulas
$sin^2x=\frac{1-cos2x}{2}$
$cos^2x=\frac{1+cos2x}{2}$
$tan^2x=\frac{1-cos2x}{1+cos2x}$

## Double-angle formulas
$sin2x = 2sinxcosx$
$cos2x = cos^2x-sin^2x = 1-2sin^2x = 2cos^2x -1$
$tan2x = \frac{2tanx}{1-tan^2x}$

### Arc
$s=r\theta$

where s denotes arc, r denotes radius, $\theta$ denotes central angle of radian

## Trigonometric sum and difference
$sin(A+B) = sinAcosB + cosAsinB$
$sin(A-B) = sinAcosB - cosAsinB$

$cos(A+B) = cosAcosB - sinAsinB$
$cos(A-B) = cosAcosB + sinAsinB$

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