Download App

Trigonometric Ratios Of Compound Angles — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Ratios Of Compound Angles4 Marks
Question
Prove that:
$\tan82\frac{1^\circ}{2}=(\sqrt{3}+\sqrt{2})(\sqrt{2}+1)=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$
✓

Answer

$\tan82\frac{1^\circ}{2}=\tan\Big(90-7\frac{1}{7}\Big)$
$=\cot7\frac{1^\circ}{2}$
$=\cot\text{A}$ if $\text{A}=7\frac{1^\circ}{2}$
Now,
$\cot\text{x}=\frac{\cos\text{x}}{\sin\text{x}}$
$=\frac{2\cos^2\text{x}}{1\sin\text{x}\cos\text{x}}$
$=\frac{1+\cos^2\text{x}}{\sin^2\text{x}}$
$\cot\text{x}=\frac{1+\cos15}{\sin15}$
$=\frac{1+\cos(45-30)}{\sin15}$
$\frac{1+\Big(\frac{1}{\sqrt{2}}\times\frac{\sqrt{3}}{2}+\frac{1}{\sqrt{2}}\times\frac{1}{2}\Big)}{\frac{1}{\sqrt{2}}\times\frac{\sqrt{3}}{2}-\frac{1}{\sqrt{2}}\times\frac{}{}\frac{1}{2}}$
$=\frac{2\sqrt{2}+(\sqrt{3}+1)}{\sqrt{3}-1}\times\frac{\sqrt{3}+1}{\sqrt{3}+1}$
$=\frac{2\sqrt{2}(\sqrt{3}+1)+(\sqrt{3}+1)^2}{3-1}$
$=\frac{2\sqrt{6}+2\sqrt{2}+4+2\sqrt{3}}{2}$
$\cot\text{x}=\sqrt{6}+\sqrt{2}+2+\sqrt{3}\ .....(1)$
$=\sqrt{2}+2\sqrt{6}+\sqrt{3}$
$=\sqrt{2}(1+\sqrt{2})+\sqrt{3}(\sqrt{2}+1)$
$\cot\text{x}=(\sqrt{2}+1)(\sqrt{2}+\sqrt{3})]\ .....(2)$
From equation (1) and (2)
$\tan82\frac{1^\circ}{2}=\cot7\frac{1^\circ}{2}=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$
$=(\sqrt{2}+1)(\sqrt{2}+\sqrt{3})$

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 "Solve the Following Question.(4 Marks)" from Trigonometric Ratios Of Compound Angles→Trigonometric Ratios Of Compound Angles — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

The $4^{\text {th }}$ and $7^{\text {th }}$ terms of G.P. are $\frac{1}{27}$ and $\frac{1}{729}$ respectively. Find the sum of $n$ terms of the G.P.
→
The income of a person is ₹ 300,000 in the first year and he receives an increase of ₹ 10000 to his income per year for the next 19 years. Find the total amount, he received in 20 years.
→
Five cards are drawn from form a pack of 52 cards. what is the chance that these 5 will contain:
  1. Just one ace
  2. At least one ace.
→
Evaluate the following limits: $\lim _{x \rightarrow \infty}\left(\frac{8 x^2+5 x+3}{2 x^2-7 x-5}\right)^{\frac{4 x+3}{8 x-1}}$
→
Show that the function f is not differentiable at $x = -3,$
where $f(x) = x^2 + 2$ for $x < -3$
$= 2 – 3x$ for $x \geq -3$
→
Determine the number of 5 card combination out of a deck of if there is one ace in each combination.
→
If the arcs of the same length in two circles subtend angles $65^\circ$ and $110^\circ$ at the centre, find the ratio of their radii.
→
Prove by the method of induction, for all n ∈ N.

$8+17+26+\ldots . .+(9 n-1)=\frac{n}{2}(9 n+7)$

→
Differentiate the following functions with respect to x:$\frac{\text{4x}+5\sin\text{x}}{\text{3x}+7\cos\text{x}}$
→
Ten cards numbered 1 to 10 are placed in a box, mixed up thoroughly, and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?
→