Reading sine and cosine
Draw an arm from the centre at the angle you want, measured anticlockwise from the positive x-axis. Where it meets the circle, read the point. The first number is the cosine, the second the sine. At 60° the point is (1/2, √3/2), so cos 60° = 1/2 and sin 60° = √3/2. At 180° the point is (-1, 0): cos 180° = −1 and sin 180° = 0. Print the unit circle chart and keep it beside you while this becomes automatic.
Reading tangent
Tangent is sine divided by cosine, the slope of the arm. At 45° the point is (√2/2, √2/2), so tan 45° = 1. At 60° it is √3/2 ÷ 1/2 = √3. At 30° it is 1/2 ÷ √3/2 = √3/3. At 90° and 270° the cosine is 0, the arm is vertical and tangent is undefined.
Degrees and radians
A radian is the angle whose arc on the unit circle is 1 unit long. The whole circle has circumference 2π, so a full turn is 2π radians and 180° is π. To convert, multiply degrees by π/180: 30° becomes π/6, 45° becomes π/4, 60° becomes π/3. The denominators 6, 4, 3 and 2 repeat around the circle: 150° is 5π/6, 135° is 3π/4, 120° is 2π/3.
The two special triangles
45-45-90: a right triangle with two equal legs of 1 has a hypotenuse of √2, so on the unit circle (hypotenuse 1) each leg is 1/√2 = √2/2. That gives the 45° point (√2/2, √2/2). 30-60-90: half an equilateral triangle of side 2 has sides 1, √3 and 2, so on the unit circle the legs are 1/2 and √3/2. The short leg is opposite 30°, so sin 30° = 1/2 and cos 30° = √3/2; at 60° they swap.
Signs in the four quadrants
Quadrant I (0° to 90°): cos and sin both positive. Quadrant II (90° to 180°): cos negative, sin positive, as at 135° where the point is (-√2/2, √2/2). Quadrant III (180° to 270°): both negative, as at 225°: (-√2/2, -√2/2). Quadrant IV (270° to 360°): cos positive, sin negative, as at 300°: (1/2, -√3/2). Every angle on the chart is one of the first-quadrant angles reflected, so learn five points and the signs and you have all sixteen.
Using the circle for identities
Because the point is on a circle of radius 1, x² + y² = 1, which is cos²θ + sin²θ = 1. Reflecting across the x-axis sends θ to −θ and flips the sign of sin only: sin(−θ) = −sin θ, cos(−θ) = cos θ. Rotating by 180° flips both. Rotating by 90° swaps them with a sign: sin(θ + 90°) = cos θ. Seeing these on the circle is faster than memorising them.
Practice
Print the blank chart from the unit circle page and fill it in from memory once a day for a week. Then try the reverse: given a point such as (−1/2, −√3/2), name the angle in degrees and radians. When both directions take under a minute, you know the circle.
A last check that catches most slips: cos and sin can never be bigger than 1 or smaller than −1, and at any angle one of them is at least √2/2 in size. If a value you have written breaks either rule, look again at the triangle you used.