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Unit circleTrigonometryHigh schoolabout 1:25

Find sin, cos and tan of 5π/4 on the unit circle: animated lesson

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

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 5π/4 on the unit circle.
  2. 2. Steps, part 1
    1. θ=225∘=5π4 rad\theta = 225^\circ = \frac{5\pi}{4}\ \text{rad}
      Convert radians to degrees (multiply by 180/π)
    2. Locate the terminal side: in Quadrant III
    3. θref=45∘\theta_{\text{ref}} = 45^\circ
      Reference angle (acute angle to the x-axis)
    4. (cos⁡θ,sin⁡θ)=(−22, −22)(\cos\theta, \sin\theta) = \left(-\frac{\sqrt{2}}{2},\ -\frac{\sqrt{2}}{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=1\tan\theta = \frac{\sin\theta}{\cos\theta} = 1
      Tangent = sine ÷ cosine
    2. csc⁡θ=−2,sec⁡θ=−2,cot⁡θ=1\csc\theta = -\sqrt{2},\quad \sec\theta = -\sqrt{2},\quad \cot\theta = 1
      Reciprocal functions
  4. 4. The answer
    (cos⁡θ,sin⁡θ)=(−22, −22)(\cos\theta, \sin\theta) = \left(-\frac{\sqrt{2}}{2},\ -\frac{\sqrt{2}}{2}\right)

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

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

Transcript

The problem. Here is the problem: Find sin, cos and tan of 5pi/4 on the unit circle. Unit circle values for 225° (5pi/4 rad).

Steps, part 1. Convert radians to degrees (multiply by 180/pi). 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 2 over 2, negative the square root of 2 over 2; tan theta equals 1. 225° (5pi/4 rad) ends in Quadrant III with a reference angle of 45°. sin theta equals negative the square root of 2 over 2, cos theta equals negative the square root of 2 over 2, tan theta equals 1, csc theta equals negative the square root of 2, sec theta equals negative the square root of 2, cot theta equals 1.

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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