Trigonometry worksheet for evaluating exact values of trigonometric functions.
Worksheet titled "Evaluating Each Trigonometric Function" with 18 trigonometric problems involving sine, cosine, tangent, and cotangent of various angles in degrees and radians.
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Step-by-step solution for: Evaluating Each Trigonometric Function worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Evaluating Each Trigonometric Function worksheets
Here are the step-by-step solutions for each trigonometric function on the worksheet.
1) $\sin(330^\circ)$
* Locate the angle: $330^\circ$ is in the 4th quadrant (between $270^\circ$ and $360^\circ$).
* Find the reference angle: $360^\circ - 330^\circ = 30^\circ$.
* Determine the sign: In the 4th quadrant, sine is negative.
* Calculate: $\sin(330^\circ) = -\sin(30^\circ) = -\frac{1}{2}$.
2) $\cos(330^\circ)$
* Locate the angle: $330^\circ$ is in the 4th quadrant.
* Find the reference angle: $30^\circ$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos(330^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2}$.
3) $\tan(120^\circ)$
* Locate the angle: $120^\circ$ is in the 2nd quadrant.
* Find the reference angle: $180^\circ - 120^\circ = 60^\circ$.
* Determine the sign: In the 2nd quadrant, tangent is negative.
* Calculate: $\tan(120^\circ) = -\tan(60^\circ) = -\sqrt{3}$.
4) $\cos\left(\frac{7\pi}{4}\right)$
* Locate the angle: $\frac{7\pi}{4}$ is in the 4th quadrant.
* Find the reference angle: $2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos\left(\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$.
5) $\cot(225^\circ)$
* Locate the angle: $225^\circ$ is in the 3rd quadrant.
* Find the reference angle: $225^\circ - 180^\circ = 45^\circ$.
* Determine the sign: In the 3rd quadrant, cotangent is positive.
* Calculate: $\cot(225^\circ) = \cot(45^\circ) = 1$.
6) $\sin\left(\frac{5\pi}{3}\right)$
* Locate the angle: $\frac{5\pi}{3}$ is in the 4th quadrant.
* Find the reference angle: $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$.
* Determine the sign: In the 4th quadrant, sine is negative.
* Calculate: $\sin\left(\frac{5\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$.
7) $\sin(150^\circ)$
* Locate the angle: $150^\circ$ is in the 2nd quadrant.
* Find the reference angle: $180^\circ - 150^\circ = 30^\circ$.
* Determine the sign: In the 2nd quadrant, sine is positive.
* Calculate: $\sin(150^\circ) = \sin(30^\circ) = \frac{1}{2}$.
8) $\cos(480^\circ)$
* Simplify the angle: $480^\circ - 360^\circ = 120^\circ$.
* Locate the angle: $120^\circ$ is in the 2nd quadrant.
* Find the reference angle: $180^\circ - 120^\circ = 60^\circ$.
* Determine the sign: In the 2nd quadrant, cosine is negative.
* Calculate: $\cos(480^\circ) = -\cos(60^\circ) = -\frac{1}{2}$.
9) $\tan(225^\circ)$
* Locate the angle: $225^\circ$ is in the 3rd quadrant.
* Find the reference angle: $225^\circ - 180^\circ = 45^\circ$.
* Determine the sign: In the 3rd quadrant, tangent is positive.
* Calculate: $\tan(225^\circ) = \tan(45^\circ) = 1$.
10) $\tan\left(\frac{9\pi}{4}\right)$
* Simplify the angle: $\frac{9\pi}{4} = \frac{8\pi}{4} + \frac{\pi}{4} = 2\pi + \frac{\pi}{4}$. This is coterminal with $\frac{\pi}{4}$.
* Locate the angle: $\frac{\pi}{4}$ is in the 1st quadrant.
* Calculate: $\tan\left(\frac{9\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) = 1$.
11) $\cos\left(\frac{17\pi}{6}\right)$
* Simplify the angle: $\frac{17\pi}{6} = \frac{12\pi}{6} + \frac{5\pi}{6} = 2\pi + \frac{5\pi}{6}$. This is coterminal with $\frac{5\pi}{6}$.
* Locate the angle: $\frac{5\pi}{6}$ is in the 2nd quadrant.
* Find the reference angle: $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.
* Determine the sign: In the 2nd quadrant, cosine is negative.
* Calculate: $\cos\left(\frac{17\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$.
12) $\sin\left(\frac{10\pi}{3}\right)$
* Simplify the angle: $\frac{10\pi}{3} = \frac{6\pi}{3} + \frac{4\pi}{3} = 2\pi + \frac{4\pi}{3}$. This is coterminal with $\frac{4\pi}{3}$.
* Locate the angle: $\frac{4\pi}{3}$ is in the 3rd quadrant.
* Find the reference angle: $\frac{4\pi}{3} - \pi = \frac{\pi}{3}$.
* Determine the sign: In the 3rd quadrant, sine is negative.
* Calculate: $\sin\left(\frac{10\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$.
13) $\cos(675^\circ)$
* Simplify the angle: $675^\circ - 360^\circ = 315^\circ$.
* Locate the angle: $315^\circ$ is in the 4th quadrant.
* Find the reference angle: $360^\circ - 315^\circ = 45^\circ$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos(675^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$.
14) $\tan\left(\frac{8\pi}{3}\right)$
* Simplify the angle: $\frac{8\pi}{3} = \frac{6\pi}{3} + \frac{2\pi}{3} = 2\pi + \frac{2\pi}{3}$. This is coterminal with $\frac{2\pi}{3}$.
* Locate the angle: $\frac{2\pi}{3}$ is in the 2nd quadrant.
* Find the reference angle: $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$.
* Determine the sign: In the 2nd quadrant, tangent is negative.
* Calculate: $\tan\left(\frac{8\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3}$.
15) $\sin(225^\circ)$
* Locate the angle: $225^\circ$ is in the 3rd quadrant.
* Find the reference angle: $225^\circ - 180^\circ = 45^\circ$.
* Determine the sign: In the 3rd quadrant, sine is negative.
* Calculate: $\sin(225^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$.
16) $\sin\left(-\frac{5\pi}{4}\right)$
* Locate the angle: $-\frac{5\pi}{4}$ is equivalent to rotating clockwise. This lands in the 2nd quadrant.
* Find the reference angle: The distance to the x-axis ($-\pi$ or $\pi$) is $\frac{\pi}{4}$.
* Determine the sign: In the 2nd quadrant, sine is positive.
* Calculate: $\sin\left(-\frac{5\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$.
17) $\cos\left(-\frac{13\pi}{6}\right)$
* Simplify the angle: $-\frac{13\pi}{6} = -2\pi - \frac{\pi}{6}$. This is coterminal with $-\frac{\pi}{6}$.
* Locate the angle: $-\frac{\pi}{6}$ is in the 4th quadrant.
* Find the reference angle: $\frac{\pi}{6}$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos\left(-\frac{13\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.
18) $\sin(390^\circ)$
* Simplify the angle: $390^\circ - 360^\circ = 30^\circ$.
* Locate the angle: $30^\circ$ is in the 1st quadrant.
* Calculate: $\sin(390^\circ) = \sin(30^\circ) = \frac{1}{2}$.
Final Answer:
1) $-\frac{1}{2}$
2) $\frac{\sqrt{3}}{2}$
3) $-\sqrt{3}$
4) $\frac{\sqrt{2}}{2}$
5) $1$
6) $-\frac{\sqrt{3}}{2}$
7) $\frac{1}{2}$
8) $-\frac{1}{2}$
9) $1$
10) $1$
11) $-\frac{\sqrt{3}}{2}$
12) $-\frac{\sqrt{3}}{2}$
13) $\frac{\sqrt{2}}{2}$
14) $-\sqrt{3}$
15) $-\frac{\sqrt{2}}{2}$
16) $\frac{\sqrt{2}}{2}$
17) $\frac{\sqrt{3}}{2}$
18) $\frac{1}{2}$
Step-by-Step Solutions
1) $\sin(330^\circ)$
* Locate the angle: $330^\circ$ is in the 4th quadrant (between $270^\circ$ and $360^\circ$).
* Find the reference angle: $360^\circ - 330^\circ = 30^\circ$.
* Determine the sign: In the 4th quadrant, sine is negative.
* Calculate: $\sin(330^\circ) = -\sin(30^\circ) = -\frac{1}{2}$.
2) $\cos(330^\circ)$
* Locate the angle: $330^\circ$ is in the 4th quadrant.
* Find the reference angle: $30^\circ$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos(330^\circ) = \cos(30^\circ) = \frac{\sqrt{3}}{2}$.
3) $\tan(120^\circ)$
* Locate the angle: $120^\circ$ is in the 2nd quadrant.
* Find the reference angle: $180^\circ - 120^\circ = 60^\circ$.
* Determine the sign: In the 2nd quadrant, tangent is negative.
* Calculate: $\tan(120^\circ) = -\tan(60^\circ) = -\sqrt{3}$.
4) $\cos\left(\frac{7\pi}{4}\right)$
* Locate the angle: $\frac{7\pi}{4}$ is in the 4th quadrant.
* Find the reference angle: $2\pi - \frac{7\pi}{4} = \frac{\pi}{4}$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos\left(\frac{7\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$.
5) $\cot(225^\circ)$
* Locate the angle: $225^\circ$ is in the 3rd quadrant.
* Find the reference angle: $225^\circ - 180^\circ = 45^\circ$.
* Determine the sign: In the 3rd quadrant, cotangent is positive.
* Calculate: $\cot(225^\circ) = \cot(45^\circ) = 1$.
6) $\sin\left(\frac{5\pi}{3}\right)$
* Locate the angle: $\frac{5\pi}{3}$ is in the 4th quadrant.
* Find the reference angle: $2\pi - \frac{5\pi}{3} = \frac{\pi}{3}$.
* Determine the sign: In the 4th quadrant, sine is negative.
* Calculate: $\sin\left(\frac{5\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$.
7) $\sin(150^\circ)$
* Locate the angle: $150^\circ$ is in the 2nd quadrant.
* Find the reference angle: $180^\circ - 150^\circ = 30^\circ$.
* Determine the sign: In the 2nd quadrant, sine is positive.
* Calculate: $\sin(150^\circ) = \sin(30^\circ) = \frac{1}{2}$.
8) $\cos(480^\circ)$
* Simplify the angle: $480^\circ - 360^\circ = 120^\circ$.
* Locate the angle: $120^\circ$ is in the 2nd quadrant.
* Find the reference angle: $180^\circ - 120^\circ = 60^\circ$.
* Determine the sign: In the 2nd quadrant, cosine is negative.
* Calculate: $\cos(480^\circ) = -\cos(60^\circ) = -\frac{1}{2}$.
9) $\tan(225^\circ)$
* Locate the angle: $225^\circ$ is in the 3rd quadrant.
* Find the reference angle: $225^\circ - 180^\circ = 45^\circ$.
* Determine the sign: In the 3rd quadrant, tangent is positive.
* Calculate: $\tan(225^\circ) = \tan(45^\circ) = 1$.
10) $\tan\left(\frac{9\pi}{4}\right)$
* Simplify the angle: $\frac{9\pi}{4} = \frac{8\pi}{4} + \frac{\pi}{4} = 2\pi + \frac{\pi}{4}$. This is coterminal with $\frac{\pi}{4}$.
* Locate the angle: $\frac{\pi}{4}$ is in the 1st quadrant.
* Calculate: $\tan\left(\frac{9\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) = 1$.
11) $\cos\left(\frac{17\pi}{6}\right)$
* Simplify the angle: $\frac{17\pi}{6} = \frac{12\pi}{6} + \frac{5\pi}{6} = 2\pi + \frac{5\pi}{6}$. This is coterminal with $\frac{5\pi}{6}$.
* Locate the angle: $\frac{5\pi}{6}$ is in the 2nd quadrant.
* Find the reference angle: $\pi - \frac{5\pi}{6} = \frac{\pi}{6}$.
* Determine the sign: In the 2nd quadrant, cosine is negative.
* Calculate: $\cos\left(\frac{17\pi}{6}\right) = -\cos\left(\frac{\pi}{6}\right) = -\frac{\sqrt{3}}{2}$.
12) $\sin\left(\frac{10\pi}{3}\right)$
* Simplify the angle: $\frac{10\pi}{3} = \frac{6\pi}{3} + \frac{4\pi}{3} = 2\pi + \frac{4\pi}{3}$. This is coterminal with $\frac{4\pi}{3}$.
* Locate the angle: $\frac{4\pi}{3}$ is in the 3rd quadrant.
* Find the reference angle: $\frac{4\pi}{3} - \pi = \frac{\pi}{3}$.
* Determine the sign: In the 3rd quadrant, sine is negative.
* Calculate: $\sin\left(\frac{10\pi}{3}\right) = -\sin\left(\frac{\pi}{3}\right) = -\frac{\sqrt{3}}{2}$.
13) $\cos(675^\circ)$
* Simplify the angle: $675^\circ - 360^\circ = 315^\circ$.
* Locate the angle: $315^\circ$ is in the 4th quadrant.
* Find the reference angle: $360^\circ - 315^\circ = 45^\circ$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos(675^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$.
14) $\tan\left(\frac{8\pi}{3}\right)$
* Simplify the angle: $\frac{8\pi}{3} = \frac{6\pi}{3} + \frac{2\pi}{3} = 2\pi + \frac{2\pi}{3}$. This is coterminal with $\frac{2\pi}{3}$.
* Locate the angle: $\frac{2\pi}{3}$ is in the 2nd quadrant.
* Find the reference angle: $\pi - \frac{2\pi}{3} = \frac{\pi}{3}$.
* Determine the sign: In the 2nd quadrant, tangent is negative.
* Calculate: $\tan\left(\frac{8\pi}{3}\right) = -\tan\left(\frac{\pi}{3}\right) = -\sqrt{3}$.
15) $\sin(225^\circ)$
* Locate the angle: $225^\circ$ is in the 3rd quadrant.
* Find the reference angle: $225^\circ - 180^\circ = 45^\circ$.
* Determine the sign: In the 3rd quadrant, sine is negative.
* Calculate: $\sin(225^\circ) = -\sin(45^\circ) = -\frac{\sqrt{2}}{2}$.
16) $\sin\left(-\frac{5\pi}{4}\right)$
* Locate the angle: $-\frac{5\pi}{4}$ is equivalent to rotating clockwise. This lands in the 2nd quadrant.
* Find the reference angle: The distance to the x-axis ($-\pi$ or $\pi$) is $\frac{\pi}{4}$.
* Determine the sign: In the 2nd quadrant, sine is positive.
* Calculate: $\sin\left(-\frac{5\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$.
17) $\cos\left(-\frac{13\pi}{6}\right)$
* Simplify the angle: $-\frac{13\pi}{6} = -2\pi - \frac{\pi}{6}$. This is coterminal with $-\frac{\pi}{6}$.
* Locate the angle: $-\frac{\pi}{6}$ is in the 4th quadrant.
* Find the reference angle: $\frac{\pi}{6}$.
* Determine the sign: In the 4th quadrant, cosine is positive.
* Calculate: $\cos\left(-\frac{13\pi}{6}\right) = \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}$.
18) $\sin(390^\circ)$
* Simplify the angle: $390^\circ - 360^\circ = 30^\circ$.
* Locate the angle: $30^\circ$ is in the 1st quadrant.
* Calculate: $\sin(390^\circ) = \sin(30^\circ) = \frac{1}{2}$.
Final Answer:
1) $-\frac{1}{2}$
2) $\frac{\sqrt{3}}{2}$
3) $-\sqrt{3}$
4) $\frac{\sqrt{2}}{2}$
5) $1$
6) $-\frac{\sqrt{3}}{2}$
7) $\frac{1}{2}$
8) $-\frac{1}{2}$
9) $1$
10) $1$
11) $-\frac{\sqrt{3}}{2}$
12) $-\frac{\sqrt{3}}{2}$
13) $\frac{\sqrt{2}}{2}$
14) $-\sqrt{3}$
15) $-\frac{\sqrt{2}}{2}$
16) $\frac{\sqrt{2}}{2}$
17) $\frac{\sqrt{3}}{2}$
18) $\frac{1}{2}$
Parent Tip: Review the logic above to help your child master the concept of trigonometric functions worksheet.
