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Unit circleTrigonometryHigh schoolabout 1:25

Find sin, cos and tan of 210° on the unit circle: animated lesson

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

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 210° on the unit circle.
  2. 2. Steps, part 1
    1. θ=210∘=7π6 rad\theta = 210^\circ = \frac{7\pi}{6}\ \text{rad}
      Convert degrees to radians (multiply by π/180)
    2. Locate the terminal side: in Quadrant III
    3. θref=30∘\theta_{\text{ref}} = 30^\circ
      Reference angle (acute angle to the x-axis)
    4. (cos⁡θ,sin⁡θ)=(−32, −12)(\cos\theta, \sin\theta) = \left(-\frac{\sqrt{3}}{2},\ -\frac{1}{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=33\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{\sqrt{3}}{3}
      Tangent = sine ÷ cosine
    2. csc⁡θ=−2,sec⁡θ=−233,cot⁡θ=3\csc\theta = -2,\quad \sec\theta = -\frac{2\sqrt{3}}{3},\quad \cot\theta = \sqrt{3}
      Reciprocal functions
  4. 4. The answer
    (cos⁡θ,sin⁡θ)=(−32, −12)(\cos\theta, \sin\theta) = \left(-\frac{\sqrt{3}}{2},\ -\frac{1}{2}\right)

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

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

Transcript

The problem. Here is the problem: Find sin, cos and tan of 210° on the unit circle. Unit circle values for 210°.

Steps, part 1. Convert degrees to radians (multiply by pi/180). Locate the terminal side: in Quadrant III. In Quadrant III sine and cosine are both 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 the square root of 3 over 2, negative 1 over 2); tan theta equals the square root of 3 over 3. 210° (7pi/6 rad) ends in Quadrant III with a reference angle of 30°. sin theta equals negative 1 over 2, cos theta equals negative the square root of 3 over 2, tan theta equals the square root of 3 over 3, csc theta equals negative 2, sec theta equals negative 2the square root of 3 over 3, cot theta equals the square root of 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 210° on the unit circle (Lesson)