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)

$si{n}^2\theta +co{s}^2\theta =1$

$1+ta{n}^2\theta =se{c}^2\theta$

$1+co{t}^2\theta =cs{c}^2\theta$

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

$si{n}^{2}x=\frac{1-cos2x}{2}$

$co{s}^{2}x=\frac{1+cos2x}{2}$

$ta{n}^{2}x=\frac{1-cos2x}{1+cos2x}$

Double-angle formulas

$sin2x=2sinxcosx$

$cos2x=co{s}^{2}x-si{n}^{2}x=1-2si{n}^{2}x=2co{s}^{2}x-1$

$tan2x=\frac{2tanx}{1-ta{n}^{2}x}$

Arc

$s=r\theta$

where s denotes arc, r denotes radius,

$\theta$ denotes central angle of radian

Trigonometric sum and difference

$sin(A+B)=sinAcosB+cosAsinB$

$sin(A-B)=sinAcosB-cosAsinB$

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

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

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

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