Unit Circle & Trig Values

A special angle — the values are exact unit-circle values.

135° · 3π/4

Step-by-step solution

1. 1

   Normalize the angle

   Subtract full turns until the angle lies between 0° and 360°: the coterminal angle is 135° = 3π/4.

   $$135° = \frac{3\pi}{4}$$
2. 2

   Quadrant and reference angle

   135° lies in quadrant II. The reference angle — the distance to the x-axis — is 45°.

   $$\text{reference angle} = 45^{\circ}$$
3. 3

   Values at the reference angle

   The unit circle gives the exact values for 45°.

   $$\sin 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \cos 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \tan 45^{\circ} = 1$$
4. 4

   Signs in quadrant II

   In quadrant II: sin is positive, cos is negative, tan is negative.

   $$\sin > 0, \quad \cos < 0, \quad \tan < 0$$
5. 5

   Result

   sin(135°) = √2/2, cos(135°) = −√2/2, tan(135°) = −1.

   $$\sin = \frac{\sqrt{2}}{2}, \quad \cos = -\frac{\sqrt{2}}{2}, \quad \tan = -1$$