Question

The approximate value of \(\sin ^{-1}(0.51)\) is .......
[Given \(\pi =3.1416\), \(\frac{2}{\sqrt{3}}=1.1547\)]

• A. \(0.535247\)
• B. \(0.535547\)
• C. \(0.535347\)
• D. \(0.535147\)

Hint:

Take x = 0.5 and a small increment 0.01, and use f(x+dx) is approximately f(x) + f'(x)dx.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept
Use \(f(x+\Delta x)\approx f(x)+f'(x)\Delta x\) with \(f(x)=\sin^{-1}x\).

Step 2: Choose values
Take \(x=0.5\) and \(\Delta x=0.01\). Then \(f(0.5)=\dfrac{\pi}{6}=\dfrac{3.1416}{6}=0.5236\).

Step 3: Derivative
\(f'(x)=\dfrac{1}{\sqrt{1-x^{2}}}\), so \(f'(0.5)=\dfrac{1}{\sqrt{0.75}}=\dfrac{2}{\sqrt{3}}=1.1547\).

Step 4: Calculation
\(f(0.51)\approx0.5236+1.1547\times0.01=0.5236+0.011547=0.535147\).

Final Answer:
The approximate value is 0.535147. \[ \boxed{\text{(D) }0.535147} \]