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 one half, but the tangent of theta is less than zero. 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 sign 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 sign of theta being equal to one half. Now the sine remember is my y value. So looking over here at my unit circle, I see that my sine is one half for 30 degrees, but it's also one 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 zero. 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 so that tells me that my tangent is not going to be less than zero here so this 30 degrees cannot be my answer. But looking over here in quadrant two, my sine is equal to one half when dealing with 150 degrees And my tangent here is negative root three over three, a negative value that is less than zero. So that satisfies both of my conditions here. So that tells me that theta is equal to 150 degrees or five pi over six radians. Thanks for watching and let me know if you have questions.

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

Business Calculus

12. Trigonometric Functions

Trigonometric Functions on the Unit Circle

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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

Step 1: Recall the unit circle definition of sine. The sine of an angle corresponds to the y-coordinate of the point on the unit circle at that angle. For sin(θ) = 1/2, we need to identify angles where the y-coordinate is 1/2.

Step 2: Determine the reference angle. The reference angle for sin(θ) = 1/2 is 30° because sin(30°) = 1/2.

Step 3: Consider the condition tan(θ) < 0. Tangent is negative in the second and fourth quadrants of the unit circle.

Step 4: Identify the angles in the second and fourth quadrants that have a reference angle of 30°. In the second quadrant, the angle is 150° (180° - 30°). In the fourth quadrant, the angle is 300° (360° - 30°).

Step 5: Verify the conditions. For θ = 150° and θ = 300°, sin(θ) = 1/2, and tan(θ) < 0. These angles satisfy the given conditions.

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

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Trigonometric Functions on the Unit Circle practice set

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