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

From given relation the value of β is 37.5°

Explanation:

Given as :

α and β are two acute angles of right triangle

Acute angle have measure less than 90°

Now given as :

`sin((x)/(2) + 2x)` =
`cos(2x +(3x)/(2))`

Or,
`cos(90° - ((x)/(2)+2x))` =
`cos(2x +(3x)/(2))`

SO,
`(90° - ((x)/(2)+2x))` =
`2x+(3x)/(2)`

Or, 90° =
`2x+(3x)/(2)` +
`(x)/(2)+2x`

or, 90° =
`(4x)/(2)` + 4x

Or, 90° =
`(12x)/(2)`

So, x =
`(90)/(6)` = 15°

∴
`sin((x)/(2) + 2x)` =
`sin((15)/(2) + 30)`

So,
`sin((x)/(2) + 2x)` = sin
`(75)/(2)`

∴ The value of Ф_1 =
`(75)/(2)` = 37.5°

Similarly
`cos(2x +(3x)/(2))` =
`cos(30 +(45)/(2))`

So ,The value of Ф_2 =
`(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