Evaluate the following: 1 2 ( ^ -1 5 3 ) - Vidyadip

Question

Evaluate the following:
$\tan\frac{1}{2}\Big(\cos^{-1}\frac{\sqrt5}{3}\Big)$

✓

Answer

Let $\frac{1}{2}\sin^{-1}\frac{3}{4}=\text{x}$
$ \sin^{-1}\frac{3}{4}=2\text{x}$
$ \sin2\text{x}=\frac{3}{4}$
$\cos2\text{x}=\frac{\sqrt7}{4}$
$\tan\Big(\frac{1}{2}\sin^{-1}\frac{3}{4}\Big)$
$=\tan\text{x}$
$ =\sqrt{\frac{1-\cos2\text{x}}{1+\cos2\text{x}}}$
$=\sqrt{\frac{1-\frac{\sqrt7}{4}}{1+\frac{\sqrt7}{4}}}$
$=\sqrt{\frac{4-\sqrt7}{4+\sqrt7}}$
$=\sqrt{\frac{\big(4-\sqrt7\big)\big(4-\sqrt7\big)}{\big(4+\sqrt7\big)\big(4-\sqrt7\big)}}$
$=\sqrt{\frac{\big(4-\sqrt7\big)^2}{9}}$
$ =\frac{4-\sqrt7}{3}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "3 Marks Question" from INVERSE TRIGNOMETRIC FUNCTIONS→INVERSE TRIGNOMETRIC FUNCTIONS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Matrices $A$ and $B$ will be inverse of each other only if

Solve the following system of homogeneous linear equations:

2x + 3y + 4z = 0,

2x - y + 3z = 0

Find the magnitude of $\vec{\text{a}}=\big(3\hat{\text{k}}+4\hat{\text{j}}\big)\times(\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}}).$

For what value of k is the following function continuous at x = 2?
$\text{f(x)}=\begin{cases}2\text{x}+1,&\text{if }\text{ x}<2\\\text{k},&\text{x}=2\\3\text{x}-1,&\text{x}>2\end{cases}$

If R and S are transitive relations on a set A, then prove that $\text{R}\cup\text{S}$ may not be a transitive relation on A.

Check the commutativity and associativity of the following binary operations:
'*' on Q defined by a * b = a + ab for all a, b ∈ Q.

For the principal values, evaluate the following:
$\sin^{-1}\Big(-\frac{\sqrt3}{2}\Big)+\cos^{-1}\Big(\frac{\sqrt3}{2}\Big)$

Find the mean and standard deviation of the following probability distributions:

$x_i$	-2	-1	0	1	2
$p_i$	0.1	0.2	0.4	0.2	0.1

Solve : $\tan ^{-1}(1)+\cos ^{-1}\left(-\frac{1}{2}\right)+\sin ^{-1}\left(-\frac{1}{2}\right)$

$\text{If y}=5\cos\text{x}-3\sin\text{x},\text{ prove that }\frac{\text{d}^2\text{y}}{\text{dx}^2}+\text{y}=0$