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 three over two and the sine of theta is less than zero. 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 three. 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 one first, I see that the cosine is equal to the square root of three over two for 30 degrees. But then going around my unit circle and ending up over here in quadrant four, I see that the cosine of 30 of three thirty degrees is also equal to the square root of two. 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 zero. 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 one where all of my functions are positive. So my sine can't be negative here, so 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 zero. So that tells us that our angle that satisfies both of these given conditions is 330 degrees or 11 pi over six. So theta is equal to 330 degrees. 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.
$\cos\theta=\frac{\sqrt3}{2}$ ; $\sin\theta$ < $0$

30°

60°

120°

330°

Verified step by step guidance

Step 1: Recall the unit circle properties. The cosine function, cos(θ), represents the x-coordinate of a point on the unit circle, while the sine function, sin(θ), represents the y-coordinate. The problem specifies that cos(θ) = √3/2 and sin(θ) < 0.

Step 2: Identify the angles where cos(θ) = √3/2. From the unit circle, cos(θ) = √3/2 occurs at angles 30° and 330° because these angles correspond to the same x-coordinate.

Step 3: Use the condition sin(θ) < 0 to narrow down the possibilities. Since sin(θ) represents the y-coordinate, it is negative in the third and fourth quadrants of the unit circle.

Step 4: Determine which of the two angles (30° or 330°) lies in the fourth quadrant. The angle 330° is in the fourth quadrant, where sin(θ) < 0, while 30° is in the first quadrant, where sin(θ) > 0.

Step 5: Conclude that the angle θ satisfying both conditions (cos(θ) = √3/2 and sin(θ) < 0) is 330°.

6:11m

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Struggling with Business Calculus?

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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Trigonometric Functions on the Unit Circle practice set

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