1. Explanation of the Exact Trig Ratios
For the angles of 60o and 30o (π/3 and π/6)
We start from an equilateral triangle with sides 2 units (this is just for convenience).
We then drop a perpendicular, bisecting the apex angle and the base.
30°
2 2 This then gives us the angles 60° and 30°
√3 And the base is bisected to give us a side in the right triangle of 1.
60°
By Pythagoras’ Theorem the perpendicular is √3
1 1
Now using the usual definitions for sine, cosine and tangent we get the following exact ratios from the right triangle
on the left.
θ 30° 60°
1
sinθ 3
2 2
cosθ 3 1
2 2
tanθ 1
3
3

2. Explanation of the Exact Trig Ratios
For the angle of 45o (π/4)
We start from an isosceles right triangle with sides of 1 unit.
By Pythagoras’ Theorem the hypotenuse is √2
√2 Because the triangle is isosceles the remaining angles must be 45°
1
45°
1
Now using the usual definitions for sine, cosine and tangent we get the following exact ratios from the right triangle.
θ 45°
1
sinθ 2
1
cosθ 2
tanθ 1