Finding exact value of trigonometric functions.

Question

How do you find the exact value of these trigonometric functions? My book has several listed so I’m just going to pick one in degrees and one in radians. 

Sin 45°

Sin pi/3

 

There are also cos and tan listed with their own angles. Would you solve them the same way you would with sin? 

Answer 1

Do you have a copy of the unit circle ?

If so look on the unit circle.

sin 45º=√2/2

sin ∏/3=60º

sin 60º=√3/2

Not every angle can be found on the unit circle. You would have to use addition and subtraction identities, double angle identities, or half-angle identities for some angles, depending on the angle.

If you don't have a unit circle copy, then you have to think about what you know about isosceles right triangles whose base angles are 45º and whose legs measure 1 unit (for convenience) and whose hypotenuse is therefore √2, and equilateral triangles with one angle bisected which gives you a 30-60-90 right triangle where each side measures 2 units (for convenience). 

 

Each side of the equilateral triangle is 2 units, not each side of the right triangle. The side opposite the 30 degree angle is half the hypotenuse.