Identify what angle, θ \theta θ , satisfies the following conditions. cos θ = 3 2 \cos\theta=\frac{\sqrt3}{2} cos θ = 2 3 <path d="M95,702c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z"> ; sin θ \sin\theta sin θ < 0 0 0

In this problem, we're asked to identify which angle theta satisfies the following conditions. And here we're told that the cosine of theta is equal to the square root of 3 over 2 and the sine of theta is less than 0. So we wanna look for an angle on our unit circle that satisfies both of these conditions. Now let's first look at our first condition, the cosine of theta being equal to the square root of 3. So looking at my unit circle, I know that the cosine of my angle is that x value in these coordinates. So looking in quadrant 1 first, I see that the cosine is equal to the square root of 3 over 2 for 30 degrees. But then going around my unit circle and ending up over here in quadrant 4, I see that the cosine of 30 of 330 degrees is also equal to the square root of 2. Now because of that, this is where our second condition comes in. Now our second condition is that the sine of theta must be less than 0. That tells us that the sine of our angle must be negative. Now looking at that first angle that I identified, 30 degrees, this is in quadrant 1 where all of my functions are positive. So my sine can't be negative here. So 30 degrees is out of the running. Now let's make sure that our 330 degrees does satisfy this condition. Now looking here at our sine, our sine is negative one half, which is indeed a negative number less than 0. So that tells us that our angle that satisfies both of these given conditions is 330 degrees or 11 pi over 6. So theta is equal to 330 degrees. Thanks for watching and let me know if you have questions.

0. Functions

Introduction to Trigonometric Functions

Calculus

0. Functions

Introduction to Trigonometric Functions

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Multiple Choice

Identify what angle, $\theta$ , satisfies the following conditions.
$\cos\theta=\frac{\sqrt3}{2}$ ; $\sin\theta$ < $0$

30°

60°

120°

330°

Verified step by step guidance

First, recall the unit circle and the values of cosine and sine for common angles. The cosine of an angle in the unit circle is the x-coordinate of the point where the terminal side of the angle intersects the circle.

The given condition is cos(θ) = √3/2. This value corresponds to angles where the x-coordinate is √3/2, which are typically 30° and 330° in the unit circle.

Next, consider the condition sin(θ) < 0. The sine of an angle is the y-coordinate of the point on the unit circle. A negative sine value indicates that the angle is in the fourth quadrant, where the y-coordinate is negative.

In the fourth quadrant, the angle that has a cosine of √3/2 and a negative sine is 330°. This is because 330° is the reflection of 30° across the x-axis, resulting in a negative sine value.

Therefore, the angle θ that satisfies both conditions cos(θ) = √3/2 and sin(θ) < 0 is 330°.

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Identify what angle, θ\thetaθ , satisfies the following conditions.cosθ=32\cos\theta=\frac{\sqrt3}{2}cosθ=23; sinθ\sin\thetasinθ < 000

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Introduction to Trigonometric Functions practice set

Identify what angle, $\theta$ , satisfies the following conditions.
$\cos\theta=\frac{\sqrt3}{2}$ ; $\sin\theta$ < $0$