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Math  /  Trigonometry

QuestionFind the following trigonometric values. Express your answers exactly. cos⁡(5π3)=□sin⁡(5π3)=□+x‾+=x+\begin{array}{l} \cos \left(\frac{5 \pi}{3}\right)=\square \\ \sin \left(\frac{5 \pi}{3}\right)=\square \frac{\overline{+x}}{+=\frac{x}{+}} \end{array}

Studdy Solution

STEP 1

1. The angle 5π3\frac{5\pi}{3} is given in radians.
2. We will use the unit circle to find the exact trigonometric values.
3. The angle 5π3\frac{5\pi}{3} is in the fourth quadrant.

STEP 2

1. Convert the angle to a more familiar reference angle.
2. Determine the cosine value.
3. Determine the sine value.

STEP 3

First, convert the angle 5π3\frac{5\pi}{3} to a reference angle. The reference angle is the angle's distance from the nearest x-axis. Since 5π3\frac{5\pi}{3} is more than 2π2\pi, we can subtract 2π2\pi to find its equivalent angle within the first circle:
5π3−2π=5π3−6π3=−π3 \frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3}
The reference angle is π3\frac{\pi}{3}.

STEP 4

Determine the cosine value. In the fourth quadrant, the cosine value is positive. The cosine of the reference angle π3\frac{\pi}{3} is:
cos⁡(π3)=12 \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
Thus,
cos⁡(5π3)=12 \cos\left(\frac{5\pi}{3}\right) = \frac{1}{2}

STEP 5

Determine the sine value. In the fourth quadrant, the sine value is negative. The sine of the reference angle π3\frac{\pi}{3} is:
sin⁡(π3)=32 \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}
Thus,
sin⁡(5π3)=−32 \sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}
The exact trigonometric values are:
cos⁡(5π3)=12 \cos\left(\frac{5\pi}{3}\right) = \frac{1}{2} sin⁡(5π3)=−32 \sin\left(\frac{5\pi}{3}\right) = -\frac{\sqrt{3}}{2}

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