# Trigonometry

## Trigonometry

• $\sin ^{2}(u)+\cos ^{2}(u)=1\,$
• $e^{{i\theta }}=\cos(\theta )+i\sin(\theta )\,$

• ${\mathrm {Real}}\left[\exp(\pm i\theta )\right]=\cos(\theta )\,$

• $\sin(x)=\sum _{{k=0}}^{\infty }{\frac {(-1)^{k}x^{{2k+1}}}{(2k+1)!}}\,$

• $\cos(x)=\sum _{{k=0}}^{\infty }{\frac {(-1)^{k}x^{{2k}}}{(2k)!}}\,$

TRIGONOMETRY BOOKS

solution Invert the matrix ${\begin{bmatrix}\cos &\sin \\-\sin &\cos \\\end{bmatrix}}$

solution Find the amplitude and period of $4\sin({\frac {1}{3}}x)\,$

solution If $\sin(x)={\frac {-4}{5}}\,$ and $x\,$ is in the third quadrant, then find $\cos(x)\,$.

solution If a 10 foot tall ladder leans against a wall, and the base of the ladder is 5 feet away from the wall, then how far up the wall does the ladder go?

solution If an 11 foot tall ladder leans against a wall, and the base of the ladder makes a 24 degree angle with the ground, then how far up the wall does the ladder go?

solution If a 30 foot tall ladder leans against a wall, and the base of the ladder makes a 65 degree angle with the ground, then how far up the wall does the ladder go?

### Basics

solution If $\cos \theta +\sin \theta ={\sqrt {2}}\cos \theta \,$ prove that $\cos \theta -\sin \theta ={\sqrt {2}}\sin \theta \,$

solution Show that ${\frac {1+\sin A-\cos A}{1+\sin A+\cos A}}+{\frac {1+\sin A+\cos A}{1+\sin A-\cos A}}=2\csc A\,$

solution Prove that $\cot ^{2}\theta [{\frac {\sec \theta -1}{1+\sin \theta }}]+\sec ^{2}\theta [{\frac {\sin \theta -1}{1+\sec \theta }}]=0\,$

solution If $\tan \theta +\sin \theta =m,\tan \theta -\sin \theta =n\,$ show that $m^{2}-n^{2}=4{\sqrt {mn}}\,$

solution Eliminate$\theta \,$ from $a\cos \theta +b\sin \theta +c=0\,$ and $a_{1}\cos \theta +b_{1}\sin \theta +c_{1}=0\,$

solution If ${\frac {x}{a}}\sin \theta +{\frac {y}{b}}\cos \theta =1\,$ and ${\frac {x}{a}}\cos \theta -{\frac {y}{b}}\sin \theta =1\,$ show that ${\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=2\,$

solution Prove that ${\frac {\csc A}{\csc A-1}}+{\frac {\csc A}{\csc A+1}}=2\sec ^{2}A\,$

solution Simplify ${\sqrt {{\frac {1+\sin A}{1-\sin A}}}}=\sec A+\tan A\,$

solution Prove that $\sin ^{6}\theta +\cos ^{6}\theta =1-3\sin ^{2}\theta \cdot \cos ^{2}\theta \,$

solution Simplify ${\frac {\sin ^{3}\theta +\cos ^{3}\theta }{\sin \theta +\cos \theta }}\,$

solution Prove that ${\frac {\sin ^{4}\theta -\cos ^{4}\theta }{\sin ^{2}\theta -\cos ^{2}\theta }}=1\,$

solution Prove that ${\frac {\tan ^{3}\theta -1}{\tan \theta -1}}=\sec ^{2}\theta +\tan \theta \,$

solution Show that ${\frac {1+\sin \theta }{\cos \theta }}+{\frac {\cos \theta }{1+\sin \theta }}=2\sec \theta \,$

solution Prove that ${\frac {\tan A}{\sec A-1}}+{\frac {\tan A}{\sec A+1}}=2\csc A\,$

TRIGONOMETRY BOOKS

## Compound Angles

$\sin(A\pm B)=\sin A\cos B\pm \cos A\sin B\,$

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

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

$\tan(A\pm B)={\frac {\tan A\pm \tan B}{1\mp \tan A\tan B}}\,$

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

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

solution Prove that $\sin(A+B)\sin(A-B)=\sin ^{2}A-\sin ^{2}B\,$ and $\cos(A+B)\cos(A-B)=\cos ^{2}A-\sin ^{2}B,\,$

solution Find the value of $\tan[{\frac {\pi }{4}}+A]\,$

solution Find the value of $\cos 105^{\circ },\sin 75^{\circ }\,$

solution what is the value of ${\frac {\tan 40+\tan 20}{\cot 45-\cot 50\cot 70}}\,$

solution Show that $\cos 40+\cos 80+\cos 160=0\,$

solution Prove that $\tan 50=\tan 40+2\tan 10\,$

solution Prove that $\cos ^{2}A+\cos ^{2}B-2\cos A\cos B\cos(A+B)=\sin ^{2}(A+B)\,$

solution Prove that $\sin ^{2}\theta +\sin ^{2}(\theta +60)+\sin ^{2}(\theta -60)={\frac {3}{2}}\,$

solution IF ${\frac {m+1}{m-1}}={\frac {\cos(\alpha -\beta )}{\sin(\alpha +\beta )}}\,$ then prove that $m=\tan[{\frac {\pi }{4}}+\alpha ]\tan[{\frac {\pi }{4}}+\beta ]\,$

solution In a triangle ABC if $\cot A+\cot B+\cot C={\sqrt {3}}\,$ then show that the triangle is equilateral.

solution $A+B+C=180^{\circ }\,$, prove that $\tan A+\tan B+\tan C=\tan A\tan B\tan C\,$

solution If $\tan \beta ={\frac {n\tan \alpha }{1+(1-n)\tan ^{2}\alpha }}\,$ then show that $\tan(\alpha -\beta )=(1-n)\tan \alpha \,$

solution If A+B=45, prove that $(1+\tan A)(1+\tan B)=2\,$.Hence show that $\tan {\frac {45}{2}}={\sqrt {2}}-1\,$

solution Prove that $\tan(A-B)+\tan(B-C)+\tan(C-A)=\tan(A-B)\tan(B-C)\tan(C-A)\,$

solution Prove that$\tan(\theta -{\frac {3\pi }{4}})\tan({\frac {7\pi }{4}}+\theta )+1=0\,$

solution Show that $\cos ^{2}\theta +\cos ^{2}(60+\theta )+\cos ^{2}(60-\theta )={\frac {3}{2}}\,$

solution Show that $\cos A+\cos(240-A)+\cos(240+A)=0\,$

## Multiple and Submultiple angles

1. $\sin 2A=2\sin A\cos A,\sin A=2\sin {\frac {A}{2}}\cos {\frac {A}{2}}\,$

2. $\sin 2A={\frac {2\tan A}{1+\tan ^{2}A}},\sin A={\frac {2\tan {\frac {A}{2}}}{1+\tan ^{2}{\frac {A}{2}}}}\,$

3. $\sin 3A=3\sin A-4\sin ^{3}A\,$

4. $\cos 2A=\cos ^{2}A-\sin ^{2}A=2\cos ^{2}A-1=1-2\sin ^{2}A\,$

5. $\cos A=\cos ^{2}{\frac {A}{2}}-\sin ^{2}{\frac {A}{2}}=2\cos ^{2}{\frac {A}{2}}-1=1-2\sin ^{2}{\frac {A}{2}}\,$

6. $\cos 2A={\frac {1-\tan ^{2}A}{1+\tan ^{2}A}},\cos A={\frac {1-\tan ^{2}{\frac {A}{2}}}{1+\tan ^{2}{\frac {A}{2}}}}\,$

TRIGONOMETRY BOOKS

7. $\cos 3A=4\cos ^{3}A-3\cos A\,$

8. $\tan 2A={\frac {2\tan ^{2}A}{1-\tan ^{2}A}},\tan A={\frac {2\tan {\frac {A}{2}}}{1-\tan ^{2}{\frac {A}{2}}}}\,$

9. $\tan 3A={\frac {3\tan A-\tan ^{3}A}{1-3\tan ^{2}A}}\,$

solution Prove that ${\frac {\cos 3A+\sin 3A}{\cos A-\sin A}}=1+2\sin 2A\,$

solution Show that $\cos ^{6}A-\sin ^{6}A=\cos 2A[1-{\frac {\sin ^{2}2A}{4}}]\,$

solution Prove that $\cot({\frac {\pi }{4}}-\theta )={\frac {\cos 2\theta }{1-\sin 2\theta }}\,$. Hence find the value of $\cot 15^{\circ }\,$

solutionIf $\tan A={\frac {1-\cos B}{\sin B}}\,$,then prove that $\tan 2A=\tan B\,$

solution Prove that $\cos({\frac {\pi }{11}})\cos({\frac {2\pi }{11}})\cos({\frac {3\pi }{11}})\cos({\frac {4\pi }{11}})\cos({\frac {5\pi }{11}})={\frac {1}{32}}\,$

solution Prove that $[1+\cos {\frac {\pi }{8}}][1+\cos {\frac {3\pi }{8}}][1+\cos {\frac {5\pi }{8}}][1+\cos {\frac {7\pi }{8}}]={\frac {1}{8}}\,$

solution Prove that $\sin A\sin[{\frac {\pi }{3}}+A]\sin[{\frac {\pi }{3}}-A]={\frac {1}{4}}\sin 3A\,$.Hence show that $\sin {\frac {\pi }{9}}\sin {\frac {2\pi }{9}}\sin {\frac {3\pi }{9}}\sin {\frac {4\pi }{9}}={\frac {3}{16}}\,$

solution Prove that $16\cos ^{5}\theta -20\cos ^{3}\theta +5\cos \theta =\cos 5\theta \,$

solution If $m\tan(\theta -30)=n\tan(\theta +120)\,$ show that $\cos 2\theta ={\frac {m+n}{2(m-n)}}\,$

solution Prove that $\sin ^{4}{\frac {\pi }{8}}+\sin ^{4}{\frac {3\pi }{8}}+\sin ^{4}{\frac {5\pi }{8}}+\sin ^{4}{\frac {7\pi }{8}}={\frac {3}{2}}\,$

solution Prove that $\cot \theta +\cot(60+\theta )-\cot(60-\theta )=3\cot 3\theta \,$

## Transformations

For all $C,D\in R\,$

1. $\sin C+\sin D=2\sin {\frac {C+D}{2}}\cos {\frac {C-D}{2}}\,$

2. $\sin C-\sin D=2\cos {\frac {C+D}{2}}\sin {\frac {C-D}{2}}\,$

3. $\cos C+\cos D=2\cos {\frac {C+D}{2}}\cos {\frac {C-D}{2}}\,$

4. $\cos C-\cos D=-2\sin {\frac {C+D}{2}}\sin {\frac {C-D}{2}}\,$

5. $2\sin A\cos B=\sin(A+B)+\sin(A-B)\,$

6. $2\cos A\sin B=\sin(A+B)-\sin(A-B)\,$

7. $2\cos A\cos B=\cos(A+B)+\cos(A-B)\,$

8.$-2\sin A\sin B=\cos(A+B)-\cos(A-B)\,$

TRIGONOMETRY BOOKS

### Inverse Trigonometric Functions

find the value of  sec x . cosx + sin ^2 ( x ) + cos ^2( x )