Question

Angles α and β are the two acute angles in a right triangle. Use the relationship between sine and cosine to find the value of β if β < α. sin( x/2 + 2x) = cos(2x + 3x/2 )
A) 15°

B) 37.5°

C) 52.5°

D) 75°

Answer 1

Answer:

From given relation the value of β is 37.5°

Step-by-step explanation:

Given as :

α and β are two acute angles of right triangle

Acute angle have measure less than 90°

Now given as :

$sin(\frac{x}{2} + 2x)$ = $cos(2x +\frac{3x}{2})$

Or, $cos(90° - (\frac{x}{2}+2x))$ =  $cos(2x +\frac{3x}{2})$

SO, $(90° - (\frac{x}{2}+2x))$ = $2x+\frac{3x}{2}$

Or, 90° =  $2x+\frac{3x}{2}$ + $\frac{x}{2}+2x$

or, 90° = $\frac{4x}{2}$ + 4x

Or,  90° =  $\frac{12x}{2}$

So, x =  $\frac{90}{6}$ = 15°

∴ $sin(\frac{x}{2} + 2x)$ = $sin(\frac{15}{2} + 30)$

So, $sin(\frac{x}{2} + 2x)$ = sin$\frac{75}{2}$

∴  The value of Ф_1 = $\frac{75}{2}$ = 37.5°

Similarly  $cos(2x +\frac{3x}{2})$ =  $cos(30 +\frac{45}{2})$

So ,The value of Ф_2 = $\frac{105}{2}$ = 52.5°

∵ β   α

So, As 37.5°52.5°

∴ β = 37.5°

Hence From given relation the value of β is 37.5°  Answer