ACT Preparation / Unit 14 ACT 1326 worked example, 60 to 90 seconds
Exact Values on the Unit Circle
The idea
Exact trig values for the special angles come from the special right triangles, in three steps. First, find the reference angle: the acute angle between the terminal side and the x-axis. Second, take the reference angle's sine and cosine from the 30-60-90 or 45-45-90 triangle with hypotenuse 1. Third, attach the signs of the quadrant: x for cosine, y for sine.
In practice
Worked example: sin 30° = 1/2, cos 30° = √3/2, and sin 45° = cos 45° = √2/2. For 150°: the reference angle is 30°, and 150° lies in the second quadrant, where x is negative, so cos 150° = −√3/2 and sin 150° = 1/2. For 225°: the reference angle is 45°, in the third quadrant, where both are negative: cos 225° = sin 225° = −√2/2.
Where people slip
The reference angle is always measured to the x-axis, never the y-axis: for 150°, it is 180 − 150 = 30°, not 60°.
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