Identify what angle, θ \theta θ , satisfies the following conditions. sin θ = 1 2 \sin\theta=\frac12 sin θ = 2 1 ; tan θ \tan\theta tan θ < 0 0 0

In this problem, we're asked to identify what angle theta satisfies the following conditions. And here, we're told that the sine of theta is equal to 1 half, but the tangent of theta is less than 0. Now this is a specific type of problem where we have to think about the value of a trig function and we also have to think about the sine of another trig function. So we need to look at our trig values on our unit circle, but we also need to pay attention to what quadrant we're in. So let's go ahead and take a look at this problem. Now looking at this first condition, the sine of theta being equal to 1 half. Now the sine, remember, is my y value, so looking over here at my unit circle, I see that my sine is 1 half for 30 degrees, but it's also 1 half for 150 degrees. And that's where this second condition comes in. So So it needs to satisfy not one but both of these conditions. Now the second condition is that the tangent of theta is less than 0. That tells us that our tangent should be negative. Now looking at our unit circle, I know in this first quadrant, all of my trig functions are positive. So that tells me that my tangent is not going to be less than 0 here, so this 30 degrees cannot be my answer. But looking over here in quadrant 2, my sine is equal to 1 half when dealing with 150 degrees, and my tangent here is negative root 3 over 3, a negative value that is less than 0. So that satisfies both of my conditions here. So that tells me that beta is equal to 150 degrees or 5 pi over 6 radians. Thanks for watching and let me know if you have questions.

3. Unit Circle

Reference Angles

Trigonometry

3. Unit Circle

Reference Angles

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Multiple Choice

Identify what angle, $\theta$ , satisfies the following conditions.
$\sin\theta=\frac12$ ; $\tan\theta$ < $0$

30°

150°

60°

300°

Verified step by step guidance

Recognize that the equation $ \sin \theta = \frac{1}{2} $ suggests that $ \theta $ could be one of the standard angles where the sine value is $ \frac{1}{2} $. These angles are typically 30° and 150° in the unit circle.

Recall that the sine function is positive in the first and second quadrants. Therefore, the angles 30° and 150° both satisfy $ \sin \theta = \frac{1}{2} $.

Consider the additional condition $ \tan \theta < 0 $. The tangent function is negative in the second and fourth quadrants.

Since 30° is in the first quadrant where tangent is positive, it does not satisfy $ \tan \theta < 0 $.

150° is in the second quadrant where tangent is negative, thus satisfying both conditions $ \sin \theta = \frac{1}{2} $ and $ \tan \theta < 0 $. Therefore, $ \theta = 150° $ is the correct angle.