Given that $sin\ \theta = \frac{a}{b}$, then $cos\ \theta$ is equal to
(A) $ \frac{b}{\sqrt{b^{2}-a^{2}}} $
(B) $ \frac{b}{a} $
(C) $ \frac{\sqrt{b^{2}-a^{2}}}{b} $
(D) $ \frac{a}{\sqrt{b^{2}-a^{2}}} $
Given:
$sin\ \theta = \frac{a}{b}$
To do:
We have to find the value of $cos\ \theta$.
Solution:
We know that,
$sin^2 \theta+cos^2 \theta=1$
$cos \theta=\sqrt{1-sin^2 \theta}$
Therefore,
$cos \theta= \sqrt{1-(\frac{a}{b})^2}$
$=\sqrt{1-\frac{a^2}{b^2}}$
$=\sqrt{\frac{b^2-a^2}{b^2}}$
$=\frac{\sqrt{b^2-a^2}}{b}$
Related Articles
- If \( tan \theta = \frac{a}{b} \), prove that\( \frac{a \sin \theta-b \cos \theta}{a \sin \theta+b \cos \theta}=\frac{a^{2}-b^{2}}{a^{2}+b^{2}} \).
- If $\frac{x}{a}cos\theta+\frac{y}{b}sin\theta=1$ and $\frac{x}{a}sin\theta-\frac{y}{b}cos\theta=1$, prove that $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=2$.
- Show that \( \frac{a \sqrt{b}-b \sqrt{a}}{a \sqrt{b}+b \sqrt{a}}=\frac{1}{a-b}(a+b-2 \sqrt{a b}) \)
- If \( x=\frac{\sqrt{a^{2}+b^{2}}+\sqrt{a^{2}-b^{2}}}{\sqrt{a^{2}+b^{2}}-\sqrt{a^{2}-b^{2}}} \), then prove that \( b^{2} x^{2}-2 a^{2} x+b^{2}=0 \).
- $\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}+\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}+\sqrt{2}}=a+b \sqrt{10}$, find a and b.
- Let $a$ and $b$ be positive integers. Show that $\sqrt{2}$ is always between $\frac{2}{b}$ and $\frac{a+2 b}{a+b}$.
- Prove that \( \sin \left(C+\frac{A+B}{2}\right)=\sin \frac{A+B}{2} \)
- Find acute angles \( A \) and \( B \), if \( \sin (A+2 B)=\frac{\sqrt{3}}{2} \) and \( \cos (A+4 B)=0, A>B \).
- Prove that \( \frac{a^{-1}}{a^{-1}+b^{-1}}+\frac{a^{-1}}{a^{-1}-b^{-1}}=\frac{2 b^{2}}{b^{2}-a^{2}} \)
- If \( x=a \sec \theta \cos \phi, y=b \sec \theta \sin \phi \) and \( z=c \tan \theta \), show that \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1 \)
- If \( \tan \theta=1 \) and \( \sin \phi =\frac{1}{\sqrt{2}}, \) then the value of \( \cos (\theta+\phi) \) is:(a) -1(b) 0(c) 1(d) \( \frac{\sqrt{3}}{2} \)
- If \( a \cos \theta+b \sin \theta=m \) and \( a \sin \theta-b \cos \theta=n \), prove that \( a^{2}+b^{2}=m^{2}+n^{2} \)
- In each of the following determine rational numbers $a$ and $b$:\( \frac{4+\sqrt{2}}{2+\sqrt{2}}=a-\sqrt{b} \)
- In each of the following determine rational numbers $a$ and $b$:\( \frac{3+\sqrt{2}}{3-\sqrt{2}}=a+b \sqrt{2} \)
- Solve: \( \frac{\sqrt{x+a}+\sqrt{x-b}}{\sqrt{x+a}-\sqrt{x-b}}=\frac{a+b}{a-b}(a \eq b) \)
Kickstart Your Career
Get certified by completing the course
Get Started
Data Structure
Networking
RDBMS
Operating System
Java
MS Excel
iOS
HTML
CSS
Android
Python
C Programming
C++
C#
MongoDB
MySQL
Javascript
PHP
Physics
Chemistry
Biology
Mathematics
English
Economics
Psychology
Social Studies
Fashion Studies
Legal Studies