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Unit circleTrigonometryHigh schoolabout 1:25

Find sin, cos and tan of 2π/3 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 2π/3 on the unit circle.
  2. 2. Steps, part 1
    1. θ=120∘=2π3 rad\theta = 120^\circ = \frac{2\pi}{3}\ \text{rad}
      Convert radians to degrees (multiply by 180/π)
    2. Locate the terminal side: in Quadrant II
    3. θref=60∘\theta_{\text{ref}} = 60^\circ
      Reference angle (acute angle to the x-axis)
    4. (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
  3. 3. Steps, part 2
    1. tan⁡θ=sin⁡θcos⁡θ=−3\tan\theta = \frac{\sin\theta}{\cos\theta} = -\sqrt{3}
      Tangent = sine ÷ cosine
    2. 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 2pi/3 on the unit circle. Unit circle values for 120° (2pi/3 rad).

Steps, part 1. Convert radians to degrees (multiply by 180/pi). Locate the terminal side: in Quadrant II. In Quadrant II sine is positive and cosine is negative (“All Students Take Calculus”). Reference angle (acute angle to the x-axis). Read the exact coordinates from the reference angle and apply the quadrant signs.

Steps, part 2. Tangent equals sine divided by cosine. Reciprocal functions.

The answer. So the answer is (cos theta, sin theta) equals (negative 1 over 2, the square root of 3 over 2; tan theta equals negative the square root of 3. 120° (2pi/3 rad) ends in Quadrant II with a reference angle of 60°. sin theta equals the square root of 3 over 2, cos theta equals negative 1 over 2, tan theta equals negative the square root of 3, csc theta equals 2the square root of 3 over 3, sec theta equals negative 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 2π/3 on the unit circle (Lesson)