Trigonometric Functions and the Unit Circle

Outcome 3: Trigonometric Functions · Blueprint Pillar 3 · PGCC MAT-1350 (interim) · Download .docx

Objectives

Define the six trigonometric functions using the unit circle.
Evaluate exact trig values at 0, π/6, π/4, π/3, π/2 and their equivalents in all four quadrants.
Convert between degrees and radians.
Use SOHCAHTOA to find unknown sides or angles in a right triangle.
Identify the period and amplitude of sine and cosine functions.

Key terms

unit circle: A circle x² + y² = 1 centered at the origin; every point is (cosθ, sinθ) for angle θ.
radian: Angle unit defined by arc length = radius; 2π radians = 360°; conversion: degrees × π/180.
sine: y-coordinate on the unit circle at angle θ; opposite/hypotenuse in a right triangle.
cosine: x-coordinate on the unit circle at angle θ; adjacent/hypotenuse in a right triangle.
tangent: sinθ/cosθ; opposite/adjacent; undefined when cosθ = 0.
reference angle: The positive acute angle between the terminal side and the x-axis; used to evaluate trig in any quadrant.
ASTC rule: All Students Take Calculus — quadrant sign rule: Q1 all positive, Q2 sin positive, Q3 tan positive, Q4 cos positive.
period: The length of one complete cycle; sin and cos have period 2π; tan has period π.
amplitude: Half the vertical range of sin or cos; |A| in f(x) = A sin(Bx + C) + D.

The concept

Trigonometry extends the right-triangle ratios to any angle — including angles greater than 90° and negative angles — by using the unit circle as the reference. This is the conceptual shift that unlocks trigonometry: move from triangles to the full plane.

The unit circle has radius 1 and center at the origin. Every angle θ measured counterclockwise from the positive x-axis corresponds to a point (x, y) on the circle. By definition, cos θ = x and sin θ = y. Immediately, since the point is on x² + y² = 1, we get the Pythagorean identity: sin²θ + cos²θ = 1 — a result that follows directly from the geometry.

Five special angles — 0, 30°, 45°, 60°, 90° (or 0, π/6, π/4, π/3, π/2 in radians) — should be memorized exactly. At 45° (π/4): both coordinates are √2/2. At 60° (π/3): sin = √3/2, cos = 1/2. At 30° (π/6): sin = 1/2, cos = √3/2. A memory pattern: sin(0°, 30°, 45°, 60°, 90°) = √(0/4), √(1/4), √(2/4), √(3/4), √(4/4) = 0, 1/2, √2/2, √3/2, 1.

To find values in other quadrants, use reference angles. The reference angle is the acute angle between the terminal side and the x-axis. The magnitude of the trig function is the same as for the reference angle; the sign depends on the quadrant. ASTC: All positive in Q1, Sine positive in Q2, Tangent positive in Q3, Cosine positive in Q4.

For right triangles, SOHCAHTOA applies: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Given one angle and one side, find any unknown side. Given two sides, find an angle using inverse trig: θ = arcsin(opposite/hypotenuse).

Blueprint Pillar 3 — Technology and Society: Trigonometry was first developed by ancient astronomers to calculate star positions and planetary orbits. Today it underpins GPS satellites (triangulation), audio engineering (Fourier transforms of sound waves), computer graphics (rotation matrices), robotics (joint angles), and civil engineering (force resolution in structures). The unit circle is not an abstract mathematical object — it is the coordinate language of every rotation and oscillation in the physical world.

Worked examples

Example 1: Finding an exact value in Q2: evaluate sin(120°). Reference angle: 180° − 120° = 60°. sin(60°) = √3/2. Q2: sine is positive. Answer: sin(120°) = √3/2. Check: the point at 120° on the unit circle is (−1/2, √3/2). y-coordinate = √3/2. ✓

Example 2: Right triangle: a building casts a shadow of 40 feet when the sun's angle of elevation is 35°. Find the building height h. tan(35°) = h/40. h = 40 · tan(35°) ≈ 40 × 0.700 ≈ 28 feet.

Common mistakes

Memorizing only Q1 values and not applying reference angles to other quadrants. At 150°, the reference angle is 30°, sin = 1/2 (positive in Q2), cos = −√3/2 (negative in Q2).
Confusing degrees and radians in a calculator. Many errors come from the calculator being in degree mode when the problem uses radians (or vice versa). Always check the mode.
Forgetting that tan is undefined at θ = π/2 + nπ. At 90°, cos = 0, and tan = sin/cos = 1/0 — undefined. The graph has a vertical asymptote, not a zero.

Self-check

Try each one before you look. A miss here costs nothing and tells you exactly what to reread.

1. What is cos(0) on the unit circle?

2. Convert 270° to radians.

3. In a right triangle, the side opposite to angle θ is 5 and the hypotenuse is 13. What is sin θ?

4. In which quadrant is sin θ positive and cos θ negative?

5. What is the period of f(x) = sin(x)?

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