Multiple Choice
Consider the equation . If is an angle in Quadrant II, what is the value of ?
3. Unit Circle
Trigonometric Functions on the Unit Circle
Multiple Choice
On the unit circle, for , when is undefined?
A
When
B
When
C
When
D
When
Verified step by step guidance1
Recall that on the unit circle, the tangent of an angle \( \theta \) is defined as the ratio of the sine to the cosine:
\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \]
Understand that tangent is undefined when the denominator of this ratio, \( \cos(\theta) \), is zero because division by zero is undefined.
Identify the values of \( \theta \) between 0 and \( \pi \) where \( \cos(\theta) = 0 \). On the unit circle, \( \cos(\theta) = 0 \) at \( \theta = \frac{\pi}{2} \).
Note that at \( \theta = \pi \), \( \cos(\pi) = -1 \), which is not zero, so tangent is defined there.
Conclude that within the interval \( 0 < \theta < \pi \), \( \tan(\theta) \) is undefined only when \( \theta = \frac{\pi}{2} \).
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