The Unit Circle and the Graphs of Sine, Cosine and Tangent
Define sine and cosine as the coordinates of a point travelling round the unit circle, which extends them beyond a right angle and explains the period, symmetry and repeated values of their graphs.
What a learner can do afterwards
- Sketch y = sin x from 0 to 2π and mark where it repeats
- Use the circle to explain why sin(π - x) = sin x
- State the period and amplitude of y = 2 sin 3x and say why tan has asymptotes
- Read sine and cosine from unit circle coordinates, for example sin 30 is 0.5, cos 60 is 0.5 and sin 90 is 1.
- Use the circle to explain why cosine turns negative in the second quadrant, for example cos 120 is -0.5 via reference 60.
1 · Read
Picture a bike wheel spinning, with a dot painted on the rim. As the wheel turns, the dot rises and falls, first fast, then slow. The unit circle captures that same rise and fall with radius 1. Each angle from the positive x axis lands the dot on a point: read x for cosine and y for sine. At 90 degrees the point is (0, 1), so sine is 1 and cosine is 0.
Let the angle grow from 0 to 2 pi and track the height. That trace is the graph of y equals sin x. It repeats every full turn, so sine and cosine have period 2 pi. That repeat is why a held musical note sounds steady, so mark where the repeat starts.
Symmetry gives mirror rules. Angles x and 180 degrees minus x share the same height, so sin of pi minus x equals sin x. Their x coordinates flip sign. In the second quadrant sine stays positive while cosine turns negative.
Take y equals 2 sin 3x. The outer 2 sets the amplitude, so the wave is twice as tall. The inner 3 squeezes three waves into one turn, so the period is 2 pi over 3. Tangent breaks where cosine is zero, at 90 and 270 degrees, and climbs between.
Coordinates give the values, the travelling point draws the graph, and symmetry explains the repeats.
The rewritten readings were written for this dataset.
More to read
Coordinates on a circle of radius 1
For a unit circle, measure θ from the positive x-axis. The point at that angle has coordinates (cos θ, sin θ). A 90° turn reaches (0, 1), giving sin 90° = 1 and cos 90° = 0.
Learn the exact coordinates for 30°, 45° and 60°, and use reflections for related angles. Cosine gives the horizontal coordinate; sine gives the vertical one. At these special angles, fractions and roots give exact values instead of decimal approximations.
From circular motion to repeating graphs
Track a point's height as its angle increases from 0 to 2π. Plotting that height against the angle produces the sine wave. The horizontal coordinate produces cosine. Both repeat after one revolution, giving a period of 2π.
Reflection explains another rule: x and 180° − x have equal vertical coordinates, so sin(180° − x) = sin x. Their horizontal coordinates have opposite signs.
Unit circle trigonometry | Lesson
Reference triangles drop from the circle point to the x axis, and their legs are |cos θ| and |sin θ|. In the second quadrant cosine turns negative while sine stays positive, giving cos 120° = −0.5 from the 60° triangle.
Always attach the quadrant sign after finding the sharp triangle value. All positive in Q1, sine only in Q2, tangent only in Q3, cosine only in Q4.
Trig unit circle review
Stretching the wave changes its look: y = 2 sin 3x has amplitude 2 (twice as tall) and period 2π/3, since the 3 squeezes three waves into one turn. Read amplitude from the multiplier, period by dividing 2π by the inner coefficient.
Tangent behaves differently, with asymptotes wherever cosine hits zero, at 90° and 270°. Between them it climbs from minus to plus infinity. Period π, breaks included.
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2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.