If + =1 3 and + =1 4 , prove that - 2 = 5 24 - Vidyadip

Question

If $\cos\alpha+\cos\beta=\frac{1}{3}$ and $\sin\alpha+\sin\beta=\frac{1}{4},$ prove that $\cos\frac{\alpha-\beta}{2}=\pm\frac{5}{24}$

✓

Answer

We have,
$\cos\alpha+\cos\beta=\frac{1}{3}$ and $\sin\alpha+\sin\beta=\frac{1}{4}$
Squaring and adding, we get
$(\cos^2\alpha+\cos^2\beta+2\cos\alpha\cos\beta)+(\sin^2\alpha+\sin^2\beta+2\sin\alpha\sin\beta)=\frac{1}{9}+\frac{1}{16}$
$\Rightarrow1+1+2(\cos\alpha\cos\beta+\sin\alpha\sin\beta)=\frac{25}{144}$
$\Rightarrow2\cos(\alpha-\beta)=\frac{25}{144}-2=\frac{-263}{144}$
$\Rightarrow\cos(\alpha-\beta)=\frac{-263}{288}$
Now,
$\cos\Big(\frac{\alpha-\beta}{2}\Big)=\sqrt{\frac{1+\cos(\alpha-\beta)}{2}}$
$=\sqrt{\frac{1-\frac{263}{288}}{2}}=\sqrt{\frac{25}{576}}$
$=\pm\frac{5}{24}$
$\therefore\cos\Big(\frac{\alpha-\beta}{2}\Big)=\pm\frac{5}{25}$

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 Trigonometric Ratios Of Compound Angles→Trigonometric Ratios Of Compound Angles — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the mean deviation about the median for the data

x_i	5	7	9	10	12	15
f_i	8	6	2	2	2	6

Find the equation of a line equidistant from the lines y = 10 and y = -2.

Show that the following four conditions are equivalent :

A $\subset$ B
A – B = $\phi$
A $\cup$ B = B
A $\cap$ B = A

Show the following quadratic equation by factorization method:
$\text{x}^2 +2\text{ x} +\frac{3}{2}=0$

$\frac{\text{a + b}}{\text{c}}=\frac{\cos\big(\frac{\text{A}-\text{B}}{2}\big)}{\sin\frac{\text{C}}{2}}$

$1+\text{i}^2+\text{i}^4+\text{i}^6+\text{i}^8+ ...+\text{i}^{20}$

Let A = {x : x $\in$ N}, B = {x : x = 2n, n $\in$ N}, C = {x : x = 2n - 1, n $\in$ N} and D = {x : x is a prime natural number}. Find:
$\text{C}\cap\text{D}$

Let a₁, a₂, ..., a_n be fixed real numbers such that f(x) = (x - a₁)(x - a₂) ...(x - a_n) What is $\lim\limits_{\text{x}\rightarrow\text{a}_1}\text{f(x)}?$ For $\text{a}\ne\text{a}_1,\text{a}_2,\dots\text{a}_\text{n}$ compute $\lim\limits_{\text{x}\rightarrow{\text{a}}}\text{f(x)}.$

Find the locus of a point which moves such that its distance from the origin is three times its distance from the x-axis.

Find (x + 1)⁶ + (x - 1)⁶. Hence or otherwise evaluate ${(\sqrt 2 + 1)^6} + {(\sqrt 2 - 1)^6}$