A figure is shown. What is the angle measure of x?

Question

Answer 1

x = 30 degrees

it's a 30-60-90 right triangle with sides 6-6sqr3- 12

30-60-90 right triangles are common right triangles in trigonometry, you'll start to recognize them

but if not, calculate x

side opposite the 30 degree smallest angle = 1/2 the hypotenuse = 1/2 tmes 12 = 6 inches

side opposite 60 degrees = 6sqr3

side opposite 90 degrees = 12

or use the pythagorean theorem

h^2 = a^2 + b^2

12^2 = 6^2 + b^2

b^2 = 144-36 = 108

b = sqr108 = sqr(3(36)) = sqr36sqr3 = 6sqr3= base

x = the angle whose tangent is opposite side/adjacent side = 6/6sqr3) = 1/sqr3 = sqr3/3

x= tan^-1(sqr3)/3 = 30 degrees

tangent of 30 degrees = (1/2)/(1/sqr3) = sqr3/2

or

sinx = 6/12 = 1/2

x = sin^-1(1/2) = 30 degrees

or use the law of sines

sinx/6 = sin90/12 = 1/12

sinx = 6/12 = 1/2

x = sin^-1(1/2) = 30 degrees

or

use the law of cosines

(sqr(144-36)^2 = 144 + 36 - 2(12)(6)Cosy where y = 90-x, solve for y, then subtract it from 90 to get x

108 = 180 -144Cosy

Cosy = -72/144 = -1/2

y = Cos^-1(1/2) = 60 degrees

90-60 = 30 degrees = x