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 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.

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0. Fundamental Concepts of Algebra3h 29m
1. Equations and Inequalities3h 27m
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3. Functions & Graphs2h 17m
4. Polynomial Functions1h 54m
5. Rational Functions1h 23m
6. Exponential and Logarithmic Functions2h 28m
7. Measuring Angles40m
8. Trigonometric Functions on Right Triangles2h 5m
9. Unit Circle1h 19m
10. Graphing Trigonometric Functions1h 19m
11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
12. Trigonometric Identities 2h 34m
13. Non-Right Triangles1h 38m
14. Vectors2h 25m
15. Polar Equations2h 5m
16. Parametric Equations1h 6m
17. Graphing Complex Numbers1h 7m
18. Systems of Equations and Matrices3h 6m
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20. Sequences, Series & Induction1h 15m
21. Combinatorics and Probability1h 45m
22. Limits & Continuity1h 49m
23. Intro to Derivatives & Area Under the Curve2h 9m

9. Unit Circle

Reference Angles

Precalculus

9. 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°

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Verified step by step guidance

Recognize that the equation $ \sin\theta = \frac{1}{2} $ implies that $ \theta $ could be an angle where the sine value is $ \frac{1}{2} $. Common angles with this sine value are 30° and 150°.

Recall that the sine function is positive in the first and second quadrants. Therefore, the angles 30° and 150° are potential solutions.

Consider the 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 $.

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Identify what angle, θ\thetaθ , satisfies the following conditions.sin⁡θ=12\sin\theta=\frac12sinθ=21​; tan⁡θ\tan\thetatanθ < 000

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