Math: Unit Circle and Trigonometry
Using angles, coordinates, and trig values on the unit circle
Math: Unit Circle and Trigonometry
Using angles, coordinates, and trig values on the unit circle
Math - Grade 9-12
- 1
On the unit circle, what point corresponds to an angle of 0 degrees?
The angle starts on the positive x-axis.
The point is (1, 0) because cosine of 0 degrees is 1 and sine of 0 degrees is 0. - 2
Find the coordinates on the unit circle for 90 degrees.
The coordinates are (0, 1) because cosine of 90 degrees is 0 and sine of 90 degrees is 1. - 3
State the exact values of sin 30 degrees and cos 30 degrees.
Use the special 30-60-90 triangle values.
The exact values are sin 30 degrees = 1/2 and cos 30 degrees = square root of 3 over 2. - 4
State the exact values of sin 45 degrees and cos 45 degrees.
The exact values are sin 45 degrees = square root of 2 over 2 and cos 45 degrees = square root of 2 over 2. - 5
Find tan 60 degrees using sin 60 degrees and cos 60 degrees.
Tangent is sine divided by cosine.
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. - 6
What are the coordinates on the unit circle for 150 degrees?
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. - 7
Find the exact value of sin 210 degrees.
First find the reference angle.
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. - 8
Find the exact value of cos 315 degrees.
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. - 9
Find the exact value of tan 135 degrees.
Use the signs of sine and cosine in Quadrant II.
The exact value is -1 because 135 degrees has a reference angle of 45 degrees and lies in Quadrant II, where tangent is negative. - 10
Convert 120 degrees to radians.
The radian measure is 2pi over 3 because 120 degrees multiplied by pi over 180 simplifies to 2pi over 3. - 11
Convert 7pi over 6 radians to degrees.
Multiply radians by 180 over pi.
The angle measure is 210 degrees because 7pi over 6 multiplied by 180 over pi equals 210. - 12
Find the reference angle for 240 degrees.
The reference angle is 60 degrees because 240 degrees is in Quadrant III, and 240 minus 180 equals 60. - 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.
Coordinates on the unit circle are written as (cos theta, sin theta).
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. - 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?
The angle is 225 degrees because both sine and cosine are negative in Quadrant III and the reference angle is 45 degrees. - 15
Explain why tan 90 degrees is undefined using unit circle values.
Look at the cosine value at 90 degrees.
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.