Trigonometry

From Exampleproblems

Jump to: navigation, search

Contents

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












Trigonometric Equations

Inverse Trigonometric Functions

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

Hyperbolic Functions

Properties of Triangles

De Moivre's Theorem

Main Page

Retrieved from "http://www.exampleproblems.com/wiki/index.php/Trigonometry"
Personal tools

Get A Wifi Network Switcher Widget for Android