Multiple Choice
Given two unit vectors and on the unit circle, what is the angle between them if = and = ? Express your answer using one significant figure.
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3. Unit Circle
Trigonometric Functions on the Unit Circle
Multiple Choice
For which values of is the expression defined on the unit circle?
A
For all such that
B
For all such that and
C
For all such that
D
For all real values of
Verified step by step guidance1
Identify the given expression: \(\sec^{2}(\theta) \cdot \cos(2\theta)\). We want to find for which values of \(\theta\) this expression is defined on the unit circle.
Recall that \(\sec(\theta) = \frac{1}{\cos(\theta)}\). Therefore, \(\sec^{2}(\theta) = \frac{1}{\cos^{2}(\theta)}\). This means the expression is undefined wherever \(\cos(\theta) = 0\) because division by zero is undefined.
Next, consider the factor \(\cos(2\theta)\). Since cosine is defined for all real numbers, \(\cos(2\theta)\) itself does not cause any undefined values. However, if \(\cos(2\theta) = 0\), the product will be zero, which is defined, so no restriction arises here.
Summarize the domain restrictions: The expression is undefined when \(\cos(\theta) = 0\). Therefore, the expression is defined for all \(\theta\) such that \(\cos(\theta) \neq 0\).
Conclude that the expression is defined for all \(\theta\) on the unit circle except where \(\cos(\theta) = 0\), which corresponds to \(\theta = \frac{\pi}{2} + k\pi\), where \(k\) is any integer.
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