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Math Grade 9-12

Math: Unit Circle and Trigonometry

Using angles, coordinates, and trig values on the unit circle

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Practice finding coordinates, sine, cosine, tangent, and angle measures using the unit circle.

Read each problem carefully. Use the unit circle relationships to find exact values when possible. Show your work in the space provided.

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Using angles, coordinates, and trig values on the unit circle

Math - Grade 9-12

Instructions: Read each problem carefully. Use the unit circle relationships to find exact values when possible. Show your work in the space provided.
  1. 1
    Unit circle with radius pointing right to the point for zero degrees.

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

  2. 2
    Unit circle with radius pointing upward to the point for ninety degrees.

    Find the coordinates on the unit circle for 90 degrees.

  3. 3
    First-quadrant unit circle radius at thirty degrees with dashed coordinate projections.

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

  4. 4
    First-quadrant unit circle radius at forty-five degrees with equal dashed projections.

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

  5. 5
    First-quadrant unit circle radius at sixty degrees with dashed component projections.

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

  6. 6
    Unit circle point in the second quadrant corresponding to one hundred fifty degrees.

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

  7. 7
    Unit circle point in the third quadrant just below the negative horizontal axis.

    Find the exact value of sin 210 degrees.

  8. 8
    Unit circle point in the fourth quadrant at a forty-five-degree reference angle.

    Find the exact value of cos 315 degrees.

  9. 9
    Unit circle point in the second quadrant at a forty-five-degree reference angle.

    Find the exact value of tan 135 degrees.

  10. 10

    Convert 120 degrees to radians.

  11. 11

    Convert 7pi over 6 radians to degrees.

  12. 12
    Third-quadrant unit circle radius with a reference angle shown to the horizontal axis.

    Find the reference angle for 240 degrees.

  13. 13
    First-quadrant unit circle point with dashed projections to show coordinate components.

    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.

  14. 14
    Unit circle point in the third quadrant with equal negative coordinate components.

    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?

  15. 15
    Unit circle at the top point showing a vertical radius and zero horizontal component.

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

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