3 · IB Maths
The unit circle
Sine and cosine are the vertical and horizontal coordinates of a point on the unit circle. Drag the point (or use the slider) and watch the wave unroll.
Predict first
Start from the default settings (θ = 45°, cos θ = 0.707). Move the angle to 135°. What does the cos θ readout show now?
135° lies in the second quadrant, where the x-coordinate is negative. The reference angle is 180° − 135° = 45°, so cos 135° = −cos 45° ≈ −0.707.
Check it in the model: Move the Angle θ slider from 45° to 135° and read the cos θ line in the readouts (0.707, then −0.707).
sin θ cos θ the point (cos θ, sin θ)
tan θ = sin θ ÷ cos θ
sin²θ + cos²θ = 1
180° = π radians
Try this
- Set θ = 30°, then 150°, 210° and 330°. Write down the sign of sin θ and cos θ in each quadrant and compare the reference angles.
- Switch the traced wave to cosine and drag the point once round. Where on the circle is cos θ = 0, and where is it at its maximum?
- Tick Show tan θ and increase θ from 80° towards 90° and beyond. What happens to the tan θ length, and why is tan 90° undefined?
Exam tip: for exact values, find the reference angle in the first quadrant, then fix the sign from the quadrant (all positive, sine, tangent, cosine going anticlockwise from the first). The readout lists the exact values at 30°, 45° and 60° and their equivalents.
More on 3: Transformations of trigonometric graphs · Solving trigonometric equations · Volume and surface area of 3D solids · Ambiguous case of the sine rule · Vector equations of lines in 3D · Compound angle and double angle identities · Sine and cosine rule · Radians, arcs and sectors · Vectors and the dot product · Voronoi diagrams