Unit Circle and Reference Angles

Develop coterminal and reference angles, coordinates and trigonometric values.

Lesson 2 of 32 parts Practice Flashcards

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Coterminal and Reference Angles

Objective: Explain coterminal and reference angles, apply the rule, and justify relevant restrictions.

Coterminal angles have the same terminal ray and differ by an integer multiple of 2π or 360°. A reference angle is the nonnegative angle between the terminal ray and the x-axis; for a nonquadrantal angle it is acute. Reduce to a standard full-turn interval before identifying the quadrant. The quadrant determines signs; the reference angle supplies the magnitude of familiar trig ratios.

Formulas

$θcoterminal=θ+2πk, k∈ℤ$

Worked examples

Try itFind the angle in [0°,360°) coterminal with 1185°.Show answer

Step 1: Subtract 3 full turns: 1185−1080=105. Step 2: Remove any remaining full turns of 360°. Step 3: The representative angle is 105°.

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Coterminal and Reference Angles

Objective: Explain coterminal and reference angles, apply the rule, and justify relevant restrictions.

Formulas

$θcoterminal=θ+2πk, k∈ℤ$

Worked examples

Try itFind the angle in [0°,360°) coterminal with 1185°.Show answer

Step 1: Subtract 3 full turns: 1185−1080=105. Step 2: Remove any remaining full turns of 360°. Step 3: The representative angle is 105°.

Coordinates and Trigonometric Values

Objective: Explain coordinates and trigonometric values, apply the rule, and justify relevant restrictions.

The unit-circle point at angle θ is (cos θ,sin θ). Because its distance from the origin is one, cos²θ+sin²θ=1. Cosine is positive on the right half-plane and sine on the upper half-plane. Tangent is y/x wherever x is nonzero. Special-angle coordinates come from 45-45-90 and 30-60-90 triangles and are scaled to hypotenuse one.

Formulas

$(x,y)=(cos θ,sin θ); tan θ=y/x$

Worked examples

Try itA unit-circle point is (7/15, √(1−(7/15)²)). What is cos θ?Show answer

Step 1: Cosine is the horizontal coordinate of the unit-circle point. Step 2: Read x=7/15. Step 3: Therefore cos θ=7/15.

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