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Unit circleTrigonometryHigh schoolabout 1:17

Find sin, cos and tan of 45° 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.

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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 (0.5 MB) · Captions (WebVTT)

Steps on screen

1. The problem
Worked example
Find sin, cos and tan of 45° on the unit circle.
2. Steps, part 1
1. $\theta = 45^\circ = \frac{\pi}{4}\ \text{rad}$
  Convert degrees to radians (multiply by π/180)
2. Locate the terminal side: in Quadrant I
3. $\theta_{\text{ref}} = 45^\circ$
  Reference angle (acute angle to the x-axis)
4. $(\cos\theta, \sin\theta) = \left(\frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2}\right)$
  Read the exact coordinates from the reference angle and apply the quadrant signs
3. Steps, part 2
1. $\tan\theta = \frac{\sin\theta}{\cos\theta} = 1$
  Tangent = sine ÷ cosine
2. $\csc\theta = \sqrt{2},\quad \sec\theta = \sqrt{2},\quad \cot\theta = 1$
  Reciprocal functions
4. The answer
$(\cos\theta, \sin\theta) = \left(\frac{\sqrt{2}}{2},\ \frac{\sqrt{2}}{2}\right)$
sin θ = √2/2, cos θ = √2/2, tan θ = 1
5. Check
- cos²θ + sin²θ = 1 (must equal 1).

Transcript

The problem. Here is the problem: Find sin, cos and tan of 45° on the unit circle. Unit circle values for 45°.

Steps, part 1. Convert degrees to radians (multiply by pi/180). Locate the terminal side: in Quadrant I. In Quadrant I sine and cosine are both positive (“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 (the square root of 2 over 2, the square root of 2 over 2; tan theta equals 1. 45° (pi/4 rad) ends in Quadrant I with a reference angle of 45°. sin theta equals the square root of 2 over 2, cos theta equals the square root of 2 over 2, tan theta equals 1, csc theta equals the square root of 2, sec theta equals 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.

Find sin, cos and tan of 45° on the unit circle (Lesson)