Question

solve for x :- cos2 30° + sin2 2x = 1​

Answer 1

Cos^2(30) + sin^2(x) =1

Sin^2(x) = 1 - cos^2(30)

sin^2(x) = sin^2(30)

Taking square root,

Sin(x) = sin(30)

=> x = nπ + (-1)^n × 30 (general solution)

=> x = 30° (definite solution)

Answer 2

Given,

$\sf cos^230+sin^2 2x=1$

We have to evaluate the value of x.

We know,

$\sf cos\: 30=\dfrac{\sqrt 3}{2}$

$\sf\implies cos^2 30=[\dfrac{\sqrt 3}{2}]^2$

$\implies\sf cos^2 30=\dfrac{3}{4}$

Applying the value to the equation,

$\sf \dfrac{3}{4}+sin^2 2x=1$

$\implies\sf sin^2 2x=1-\dfrac{3}{4}$

$\implies\sf sin^2 2x=\dfrac{1}{4}$

$\implies\sf sin 2x=\sqrt{\dfrac{1}{4}}$

$\implies\sf sin 2x={\dfrac{1}{2}}$

• The length of the shortest side is   $\bf\dfrac{1}{2}$ .

• This side is opposite the smallest angle - 30 degrees.

• The length of the hypotenuse is 1

Therefore,

$\sf sin\:30=\dfrac{1}{2}$

If x =  15 ,

$\sf sin\:2x=sin\:30=\dfrac{1}{2}$

The next angle beyond  30  degrees such that  :

$\sf sin\:x=\dfrac{1}{2}=150^0=180-30$

Therefore x = 75.

Therefore ,

x can have the values :

► x = 15 ◄   OR   ► x = 75 ◄