Trigonometry/For Enthusiasts/Less-Used Trig Identities

From Wikibooks, open books for an open world
< Trigonometry
Jump to: navigation, search

Triangle Identities[edit]

In addition to the Law of Sines, the Law of Cosines, and the Law of Tangents, there are numerous other identities that apply to the three angles A, B, and C of any triangle (where A+B+C=180° and each of A, B, and C is greater than zero). Some of the most notable ones follow:

\cos^2(A)+\cos^2(B)+\cos^2(C)+2\cos(A)\cos(B)\cos(C)=1
\sin(A)+\sin(B)+\sin(C)=4\cos\bigl(\tfrac{A}{2}\bigr)\cos\bigl(\tfrac{B}{2}\bigr)\cos\bigl(\tfrac{C}{2}\bigr)
\tan(A)+\tan(B)+\tan(C)=\tan(A)\tan(B)\tan(C)
\tan\bigl(\tfrac{A}{2}\bigr)\tan\bigl(\tfrac{B}{2}\bigr)+\tan\bigl(\tfrac{B}{2}\bigr)\tan\bigl(\tfrac{C}{2}\bigr)+\tan\bigl(\tfrac{C}{2}\bigr)\tan\bigl(\tfrac{A}{2}\bigr)=1
\cot(A)\cot(B)+\cot(B)\cot(C)+\cot(C)\cot(A)=1
\cot\bigl(\tfrac{A}{2}\bigr)\cot\bigl(\tfrac{B}{2}\bigr)\cot\bigl(\tfrac{C}{2}\bigr)=\cot\bigl(\tfrac{A}{2}\bigr)+\cot\bigl(\tfrac{B}{2}\bigr)+\cot\bigl(\tfrac{C}{2}\bigr)
\sin(A)\sin(B)\sin(C)=\frac{1}{\bigl(\cot(A)+\cot(B)\bigr)\bigl(\cot(B)+\cot(C)\bigr)\bigl(\cot(C)+\cot(A)\bigr)}
\frac{\sin(A)+\sin(B)-\sin(C)}{\sin(A)+\sin(B)+\sin(C)}=\tan\bigl(\tfrac{A}{2}\bigr)\tan\bigl(\tfrac{B}{2}\bigr)

Pythagoras[edit]

\sin^2(x)+\cos^2(x)=1
1+\tan^2(x)=\sec^2(x)
1+\cot^2(x)=\csc^2(x)

These are all direct consequences of Pythagoras's theorem.

Sum/Difference of angles[edit]

\cos(x\pm y)=\cos(x)\cos(y)\mp\sin(x)\sin(y)
\sin(x\pm y)=\sin(x)\cos(y)\pm\sin(y)\cos(x)
\tan(x\pm y)=\frac{\tan(x)\pm\tan(y)}{1\mp\tan(x)\tan(y)}

Product to Sum[edit]

2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)
2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)
2\sin(x)\cos(y)=\sin(x-y)+\sin(x+y)

Sum and difference to product[edit]

A\sin(x)+B\cos(x)=C\sin(x+y) , where C=\sqrt{A^2+B^2} and y=\pm\arctan\bigl(\tfrac{B}{A}\bigr)
\sin(\alpha)+\sin(\beta)=2\sin\bigl(\tfrac{\alpha+\beta}{2}\bigr)\cos\bigl(\tfrac{\alpha-\beta}{2}\bigr)
\sin(\alpha)-\sin(\beta)=2\cos\bigl(\tfrac{\alpha+\beta}{2}\bigr)\sin\bigl(\tfrac{\alpha-\beta}{2}\bigr)
\cos(\alpha)+\cos(\beta)=2\cos\bigl(\tfrac{\alpha+\beta}{2}\bigr)\cos\bigl(\tfrac{\alpha-\beta}{2}\bigr)
\cos(\alpha)-\cos(\beta)=-2\sin\bigl(\tfrac{\alpha+\beta}{2}\bigr)\sin\bigl(\tfrac{\alpha-\beta}{2}\bigr)

Multiple angle[edit]

\cos(2x)=\cos^2(x)- \sin^2(x)=2\cos^2(x)-1=1-2\sin^2(x)
\sin(2x)=2\sin(x)\cos(x)
\tan(2x)=\frac{2\tan(x)}{1- \tan^2(x)}
\cot(2x)=\frac{\cot(x)-\tan(x)}{2}
\csc(2x)=\frac{\cot(x)+\tan(x)}{2}
\cos(3x)=4\cos^3(x)-3\cos(x)
\sin(3x)=-4\sin^3(x)+3\sin(x)
\tan(3x)=\frac{3\tan(x)-\tan^3(x)}{1-3\tan^2(x)}
\cos(4x)=8\cos^4(x)-8\cos^2(x)+1
\sin(4x)=4\sin(x)\cos^3(x)-4\sin^3(x)\cos(x)
\sin^2(4x)=16\Big[\sin^2(x)-5\sin^4(x)+8\sin^6(x)-4\sin^8(x)\Big]
\tan(4x)=\frac{4\tan(x)-4\tan^3(x)}{1-6\tan^2(x)+\tan^4(x)}
\cos(5x)=16\cos^5(x)-20\cos^3(x)+5\cos(x)
\sin(5x)=16\sin^5(x)-20\sin^3(x)+5\sin(x)
\tan(5x)=\frac{5\tan(x)-10\tan^3(x)+\tan^5(x)}{1-10\tan^2(x)+5\tan^4(x)}
\cos(6x)=32\cos^6(x)-48\cos^4(x)+18\cos^2(x)-1
\cos(7x)=64\cos^7(x)-112\cos^5(x)+56\cos^3(x)-7\cos(x)
\sin(7x)=-64\sin^7(x)+112\sin^5(x)-56\sin^3(x)+7\sin(x)
\cos(8x)=128\cos^8(x)-256\cos^6(x)+160\cos^4(x)-32\cos^2(x)+1
\cos(nx)=2\cos(x)\cos\bigl((n-1)x\bigr)-\cos\bigl((n-2)x\bigr)
\sin(nx)=2\cos(x)\sin\bigl((n-1)x\bigr)-\sin\bigl((n-2)x\bigr)

These are all direct consequences of the sum/difference formulae

Half angle[edit]

\cos\bigl(\tfrac{x}{2}\bigr)=\pm\sqrt{\frac{1+\cos(x)}{2}}
\sin\bigl(\tfrac{x}{2}\bigr)=\pm\sqrt{\frac{1-\cos(x)}{2}}
\tan\bigl(\tfrac{x}{2}\bigr)=\frac{1-\cos(x)}{\sin(x)}=\frac{\sin(x)}{1+\cos(x)}=\pm\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}
\cos^2\bigl(\tfrac{3x}{2}\bigr)=2\cos^3(x)-\frac{3\cos(x)+1}{2}

In cases with \pm , the sign of the result must be determined from the value of \frac{x}{2} . These derive from the \cos(2x) formulae.

Power Reduction[edit]

\sin^2(\theta)=\frac{1-\cos2\theta}{2}
\cos^2(\theta)=\frac{1+\cos2\theta}{2}
\tan^2(\theta)=\frac{1-\cos2\theta}{1+\cos2\theta}

Even/Odd[edit]

\sin(-\theta)=-\sin(\theta)
\cos(-\theta)=\cos(\theta)
\tan(-\theta)=-\tan(\theta)
\csc(-\theta)=-\csc(\theta)
\sec(-\theta)=\sec(\theta)
\cot(-\theta)=-\cot(\theta)

Calculus[edit]

\frac{d}{dx}[\sin(x)]=\cos(x)
\frac{d}{dx}[\cos(x)]=-\sin(x)
\frac{d}{dx}[\tan(x)]=\sec^2(x)
\frac{d}{dx}[\sec(x)]=\sec(x)\tan(x)
\frac{d}{dx}[\csc(x)]=-\csc(x)\cot(x)
\frac{d}{dx}[\cot(x)]=-\csc^2(x)