Multiple Choice
On the unit circle, in which quadrant are both and negative?
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3. Unit Circle
Trigonometric Functions on the Unit Circle
Multiple Choice
Which of the following explains why equals using the unit circle?
A
On the unit circle, the x-coordinate at is always greater than the y-coordinate at .
B
Because and are both in the first quadrant, their cosine and sine values are always equal.
C
The unit circle shows that and are both negative, so they must be equal.
D
On the unit circle, the x-coordinate at is equal to the y-coordinate at , so equals .
Verified step by step guidance1
Recall that on the unit circle, any point corresponding to an angle \( \theta \) has coordinates \( (\cos(\theta), \sin(\theta)) \). This means the x-coordinate is \( \cos(\theta) \) and the y-coordinate is \( \sin(\theta) \).
Identify the coordinates for the angles 60° and 30° on the unit circle: the point at 60° is \( (\cos(60^\circ), \sin(60^\circ)) \) and the point at 30° is \( (\cos(30^\circ), \sin(30^\circ)) \).
Notice that \( \cos(60^\circ) \) represents the x-coordinate of the point at 60°, and \( \sin(30^\circ) \) represents the y-coordinate of the point at 30°.
Using the symmetry of the unit circle, observe that the x-coordinate at 60° is the same as the y-coordinate at 30°, because 60° and 30° are complementary angles (they add up to 90°). This is a key property: \( \cos(\theta) = \sin(90^\circ - \theta) \).
Therefore, \( \cos(60^\circ) = \sin(30^\circ) \) because the x-coordinate at 60° equals the y-coordinate at 30° on the unit circle.
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