How to find exact value of theta in degree and radian measure.

Question

My book has quite a few examples so I’ll just pick three. 

The directions are: 

find the exact value of theta in degree measure (0<theta<90°) and in radian measure (0<theta<90°), without using your GDC.

cos theta = 1/2 

sin theta = √2 /2

tan theta = √3

Answer 1

draw a right triangle for each of these

cosθ=1/2

cos=adjacent/hypotenuse

in a 30-60-90 right triangle, the side opposite the 30º angle is half the hypotenuse

the angle opposite the leg that measures 1 must be 30º

therefore θ must be 60º or ∏/3

 

sinθ=√2/2

sin=opposite/hypotenuse

using the Pythagorean Theorem...

2²=(√2)²+x²

4=2+x²

2=x²

x=√2

you have a right isosceles triangle with the base angles the same; each one measures 45º

therefore θ=45º or ∏/4

 

tan=opposite/adjacent

tanθ=√3/1

using the Pythagorean Theorem...

h²=(√3)²+1²

h²=3+1

h²=4

h=2

you have a right triangle where one leg is 1 and the hypotenuse is 2

therefore you have a 30º angle and the side opposite this angle is half the hypotenuse

the adjacent side is 1, the hypotenuse is 2, and the other leg is √3

seeing that tan=opposite/adjacent, the adjacent side is 1 so the opposite angle is 30º and θ=60º or ∏/3