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Precalculus Unit 6: Right-triangle & unit-circle trig

Trigonometry starts with angles measured in radians and the unit circle, which defines sine and cosine for any angle.

Covers: Radian measure, unit circle, reference angles, special values

Study guide

Angles can be measured in degrees or radians. A radian is the angle that cuts off an arc equal to the radius, so a full circle is 2π radians and 180° = π. Multiply by π/180 to convert degrees to radians. Arc length on a circle of radius r is s = rθ when θ is in radians.

Right-triangle trig defines sine, cosine and tangent as ratios of sides (SOH-CAH-TOA). The unit circle extends these definitions to any angle: the terminal side of angle θ meets the circle of radius 1 at (cos θ, sin θ), and tan θ = sin θ/cos θ. So sine and cosine are always between −1 and 1, and tangent is undefined where cosine is 0.

A reference angle is the acute angle between the terminal side and the x-axis. Any angle's trig values have the same size as its reference angle's, with signs set by the quadrant: in Quadrant I everything is positive, in II only sine is positive, in III only tangent, and in IV only cosine.

Know the special values exactly: sin(π/6) = 1/2, sin(π/4) = √2/2, sin(π/3) = √3/2, with cosine values in the reverse order. These come from the 30-60-90 and 45-45-90 triangles and let you find exact values for all their multiples around the circle.

If the terminal side of θ passes through (x, y), let r = √(x² + y²); then sin θ = y/r, cos θ = x/r and tan θ = y/x. This extends the unit-circle definitions to a circle of any radius. Angles that differ by a multiple of 2π (360°) are coterminal and have the same trig values. A negative angle is measured clockwise from the positive x-axis.

Key ideas

• A radian is the angle that cuts off an arc equal to the radius; 2π radians = 360°.
• On the unit circle, the point at angle θ is (cos θ, sin θ).
• tan θ = sin θ / cos θ.
• Memorize values at 0, π/6, π/4, π/3, π/2 and use reference angles and quadrant signs for the rest.
• SOH-CAH-TOA for right triangles: sin = opp/hyp, cos = adj/hyp, tan = opp/adj.

Formulas and rules

Common mistakes

• Calculator in degree mode for a radian problem.
• Wrong signs in quadrants II–IV (All Students Take Calculus).

Words to know

Degrees to radians

Multiply by π/180 (60° = π/3).

sin(π/6)

1/2

cos(π/4)

√2/2

Point on the unit circle at angle θ

(cos θ, sin θ)

Quadrant III

Sine and cosine are both negative; tangent is positive.

Arc length formula

s = rθ (θ in radians)

Radian

The angle that cuts off an arc equal to the radius.

Unit circle

Circle of radius 1; the point at angle θ is (cos θ, sin θ).

Reference angle

The acute angle between the terminal side and the x-axis.

Tangent (unit circle)

tan θ = sin θ/cos θ; undefined where cos θ = 0.

sin(π/3)

√3/2

cos(π/6)

√3/2

Special angle values

Exact sine and cosine at π/6, π/4 and π/3 from the 30-60-90 and 45-45-90 triangles.

Sample practice questions

31 more questions like these, from level 1 to 5, are in the practice set.