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Unit circleTrigonometryHigh schoolabout 1:35

Find sin, cos and tan of -60° on the unit circle: animated lesson

A narrated, step-by-step unit circle video lesson with captions. Final answer: (cos θ, sin θ) = (1/2, −√3/2); tan θ = −√3.

Video with a synthetic English voice and captions. Pick another language for subtitles in that language, or open the interactive version.

Video narrated by a synthetic voice (Piper TTS, en_US-ljspeech-high voice (LJ Speech dataset, public domain)); the steps come from our maths engine and the answer was checked independently. Download MP4 (0.6 MB) · Captions (WebVTT)

Steps on screen

  1. 1. The problem
    Worked example
    Find sin, cos and tan of -60° on the unit circle.
  2. 2. Steps, part 1
    1. θ=−60∘=−π3 rad\theta = -60^\circ = -\frac{\pi}{3}\ \text{rad}
      Convert degrees to radians (multiply by π/180)
    2. 300∘300^\circ
      Find the coterminal angle between 0° and 360°
    3. Locate the terminal side: in Quadrant IV
    4. θref=60∘\theta_{\text{ref}} = 60^\circ
      Reference angle (acute angle to the x-axis)
  3. 3. Steps, part 2
    1. (cos⁡θ,sin⁡θ)=(12, −32)(\cos\theta, \sin\theta) = \left(\frac{1}{2},\ -\frac{\sqrt{3}}{2}\right)
      Read the exact coordinates from the reference angle and apply the quadrant signs
    2. tan⁡θ=sin⁡θcos⁡θ=−3\tan\theta = \frac{\sin\theta}{\cos\theta} = -\sqrt{3}
      Tangent = sine ÷ cosine
    3. csc⁡θ=−233,sec⁡θ=2,cot⁡θ=−33\csc\theta = -\frac{2\sqrt{3}}{3},\quad \sec\theta = 2,\quad \cot\theta = -\frac{\sqrt{3}}{3}
      Reciprocal functions
  4. 4. The answer
    (cos⁡θ,sin⁡θ)=(12, −32)(\cos\theta, \sin\theta) = \left(\frac{1}{2},\ -\frac{\sqrt{3}}{2}\right)

    sin θ = −√3/2, cos θ = 1/2, tan θ = −√3

  5. 5. Check
    • cos²θ + sin²θ = 1 (must equal 1).

Transcript

The problem. Here is the problem: Find sin, cos and tan of negative 60° on the unit circle. Unit circle values for negative 60°.

Steps, part 1. Convert degrees to radians (multiply by pi/180). Find the coterminal angle between 0° and 360°. Add or subtract full turns of 360° (2pi); trigonometric values repeat every full turn. Locate the terminal side: in Quadrant IV. In Quadrant IV sine is negative and cosine is positive (“All Students Take Calculus”). Reference angle (acute angle to the x-axis).

Steps, part 2. Read the exact coordinates from the reference angle and apply the quadrant signs. Tangent equals sine divided by cosine. Reciprocal functions.

The answer. So the answer is (cos theta, sin theta) equals (1 over 2, negative the square root of 3 over 2; tan theta equals negative the square root of 3. negative 60° (-pi/3 rad) ends in Quadrant IV with a reference angle of 60°. sin theta equals negative the square root of 3 over 2, cos theta equals 1 over 2, tan theta equals negative the square root of 3, csc theta equals negative 2the square root of 3 over 3, sec theta equals 2, cot theta equals negative the square root of 3 over 3.

Check. cos squaredtheta plus sin squaredtheta equals 1 (must equal 1).

About unit circle

Read sine, cosine and tangent of standard angles from the unit circle.

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Find sin, cos and tan of -60° on the unit circle (Lesson)