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="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 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 sign, our sign 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.

3. Unit Circle

Reference Angles

Trigonometry3. Unit CircleReference Angles

Multiple Choice

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

30°

60°

120°

330°

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

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