Question:
If \( \tan \frac{\theta}{2} = \frac{1}{2} \), then the value of \( \sin \theta \) is
If \( \tan \frac{\theta}{2} = \frac{1}{2} \), then the value of \( \sin \theta \) is
Show Hint
Memorize half-angle identities to quickly convert \( \tan \frac{\theta}{2} \) into \( \sin \theta \) or \( \cos \theta \).
Updated On: May 8, 2026
- \( \frac{4}{5} \)
- \( \frac{3}{5} \)
- \( \frac{1}{2} \)
- \( 1 \)
- \( \frac{2}{5} \)
Show Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Concept:
Use the identity:
\[
\sin \theta = \frac{2\tan \frac{\theta}{2}}{1+\tan^2 \frac{\theta}{2}}
\]
Step 1: Substitute given value
\[ \tan \frac{\theta}{2} = \frac{1}{2} \]
Step 2: Apply identity
\[ \sin \theta = \frac{2 \cdot \frac{1}{2}}{1 + \left(\frac{1}{2}\right)^2} \]
Step 3: Simplify numerator
\[ 2 \cdot \frac{1}{2} = 1 \]
Step 4: Simplify denominator
\[ 1 + \frac{1}{4} = \frac{5}{4} \]
Step 5: Final simplification
\[ \sin \theta = \frac{1}{5/4} = \frac{4}{5} \]
Step 6: Final Answer
\[ \boxed{\frac{4}{5}} \]
Step 1: Substitute given value
\[ \tan \frac{\theta}{2} = \frac{1}{2} \]
Step 2: Apply identity
\[ \sin \theta = \frac{2 \cdot \frac{1}{2}}{1 + \left(\frac{1}{2}\right)^2} \]
Step 3: Simplify numerator
\[ 2 \cdot \frac{1}{2} = 1 \]
Step 4: Simplify denominator
\[ 1 + \frac{1}{4} = \frac{5}{4} \]
Step 5: Final simplification
\[ \sin \theta = \frac{1}{5/4} = \frac{4}{5} \]
Step 6: Final Answer
\[ \boxed{\frac{4}{5}} \]
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Trigonometry Questions
- \( \tan 15^\circ + \tan 75^\circ = \)
- If \( x + z = 2y \) and \( y = \frac{\pi}{4} \), then \( \tan x \tan y \tan z = \)
- If \( \sin x + \sin y = a \), \( \cos x + \cos y = b \) and \( x + y = \frac{2\pi}{3} \), then the value of \( \frac{a}{b} \) is equal to
- If \( \sin \alpha = \frac{12}{13} \), where \( \frac{\pi}{2}<\alpha<\frac{3\pi}{2} \), then the value of \( \tan \alpha \) is equal to
- If \( f(x) = \tan^{-1}\left(\frac{2x}{1 - x^2}\right) \), then \( f\left(\frac{1}{\sqrt{3}}\right) \) is equal to
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions