Sine: Value

First, find the reference angle of 3π/4.

3π/4 is between π/2 and π.
So draw the terminal side on quadrant II.

Then the reference angle is
π - 3π/4.

π = 4π/4

4π/4 - 3π/4 = π/4

So the reference angle is π/4.

Draw a right triangle like this.

The central angle is π/4: 45º.
So this is a 45-45-90 triangle.

So the base is -1.
The height is 1.
And the hypotenuse is √2.

sin 3π/4 is the sine of the right triangle.

Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (√2).

So sin 3π/4 = 1/√2.

Rationalize the denominator √2
by multiplying √2/√2.

Then √2/2.

So sin 3π/4 = √2/2.

θ = 0

The adjacent side (base) is 1.
The opposite side (height) is 0.
The hypotenuse is 1.

SOH:
Sine,
Opposite side (0),
Hypotenuse (1).

So sin 0 = 0/1 = 0.

θ = π/6

This is a 30-60-90 triangle.

The adjacent side (base) is √3.
The opposite side (height) is 1.
The hypotenuse is 2.

SOH:
Sine,
Opposite side (1),
Hypotenuse (2).

So sin π/6 = 1/2.

θ = π/4

This is a 45-45-90 triangle.

The adjacent side (base) is 1.
The opposite side (height) is 1.
The hypotenuse is √2.

SOH:
Sine,
Opposite side (1),
Hypotenuse (√2).

So sin π/4 = 1/√2 (= √2/2).

θ = π/3

This is a 30-60-90 triangle.

The adjacent side (base) is 1.
The opposite side (height) is √3.
The hypotenuse is 2.

SOH:
Sine,
Opposite side (√3),
Hypotenuse (2).

So sin π/3 = √3/2.

θ = π/2

The adjacent side (base) is 0.
The opposite side (height) is 1.
The hypotenuse is 1.

SOH:
Sine,
Opposite side (1),
Hypotenuse (1).

So sin π/2 = 1/1 = 1.