Math: Unit Circle and Trigonometry Worksheet

Question 1

On the unit circle, what point corresponds to an angle of 0 degrees?

Accepted Answer

The point is (1, 0) because cosine of 0 degrees is 1 and sine of 0 degrees is 0.

Question 2

Find the coordinates on the unit circle for 90 degrees.

Accepted Answer

The coordinates are (0, 1) because cosine of 90 degrees is 0 and sine of 90 degrees is 1.

Question 3

State the exact values of sin 30 degrees and cos 30 degrees.

Accepted Answer

The exact values are sin 30 degrees = 1/2 and cos 30 degrees = square root of 3 over 2.

Question 4

State the exact values of sin 45 degrees and cos 45 degrees.

Accepted Answer

The exact values are sin 45 degrees = square root of 2 over 2 and cos 45 degrees = square root of 2 over 2.

Question 5

Find tan 60 degrees using sin 60 degrees and cos 60 degrees.

Accepted Answer

The exact value is tan 60 degrees = square root of 3 because tan theta = sin theta divided by cos theta, and (square root of 3 over 2) divided by (1/2) = square root of 3.

Question 6

What are the coordinates on the unit circle for 150 degrees?

Accepted Answer

The coordinates are (-square root of 3 over 2, 1/2) because 150 degrees is in Quadrant II, where cosine is negative and sine is positive.

Question 7

Find the exact value of sin 210 degrees.

Accepted Answer

The exact value is -1/2 because 210 degrees has a reference angle of 30 degrees and lies in Quadrant III, where sine is negative.

Question 8

Find the exact value of cos 315 degrees.

Accepted Answer

The exact value is square root of 2 over 2 because 315 degrees has a reference angle of 45 degrees and lies in Quadrant IV, where cosine is positive.

Question 9

Find the exact value of tan 135 degrees.

Accepted Answer

The exact value is -1 because 135 degrees has a reference angle of 45 degrees and lies in Quadrant II, where tangent is negative.

Question 10

Convert 120 degrees to radians.

Accepted Answer

The radian measure is 2pi over 3 because 120 degrees multiplied by pi over 180 simplifies to 2pi over 3.

Question 11

Convert 7pi over 6 radians to degrees.

Accepted Answer

The angle measure is 210 degrees because 7pi over 6 multiplied by 180 over pi equals 210.

Question 12

Find the reference angle for 240 degrees.

Accepted Answer

The reference angle is 60 degrees because 240 degrees is in Quadrant III, and 240 minus 180 equals 60.

Question 13

On the unit circle, the point for an angle theta is (square root of 3 over 2, 1/2). Find sin theta and cos theta.

Accepted Answer

Sin theta is 1/2 and cos theta is square root of 3 over 2 because on the unit circle, the x-coordinate is cosine and the y-coordinate is sine.

Question 14

If sin theta = -square root of 2 over 2 and cos theta = -square root of 2 over 2, what angle between 0 degrees and 360 degrees is theta?

Accepted Answer

The angle is 225 degrees because both sine and cosine are negative in Quadrant III and the reference angle is 45 degrees.

Question 15

Explain why tan 90 degrees is undefined using unit circle values.

Accepted Answer

Tangent is undefined at 90 degrees because tan theta = sin theta divided by cos theta. On the unit circle, sin 90 degrees = 1 and cos 90 degrees = 0, so this gives 1 divided by 0, which is undefined.

Math: Unit Circle and Trigonometry

Math: Unit Circle and Trigonometry

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