Unit Circle and Trigonometric Ratios

Explore unit circle coordinates, reference angles and signs, right-triangle ratios.

Lesson 2 of 53 parts Practice Flashcards

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Unit Circle Coordinates

Objective: Explain unit circle coordinates and use it to solve or interpret a problem.

At angle θ, the unit-circle point is (cosθ,sinθ). Signs depend on quadrant, and tanθ=sinθ/cosθ when cosine is nonzero. Cosine can equal zero in its denominator.

Worked examples

Try itUnit Circle and Trigonometric Ratios — practice 1.1: Find sin(π/2).Show answer

Step 1: Locate π/2 on the unit circle. Step 2: Read the y-coordinate of that point. Step 3: The sine value is 1.

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Unit Circle Coordinates

Objective: Explain unit circle coordinates and use it to solve or interpret a problem.

At angle θ, the unit-circle point is (cosθ,sinθ). Signs depend on quadrant, and tanθ=sinθ/cosθ when cosine is nonzero. Cosine can equal zero in its denominator.

Worked examples

Try itUnit Circle and Trigonometric Ratios — practice 1.1: Find sin(π/2).Show answer

Step 1: Locate π/2 on the unit circle. Step 2: Read the y-coordinate of that point. Step 3: The sine value is 1.

Reference Angles and Signs

Objective: Explain reference angles and signs and use it to solve or interpret a problem.

The acute reference angle provides absolute trig values; quadrant determines each sign. Do not apply a positive reference value without restoring its sign. Negative, because x-coordinates there are negative.

Worked examples

Try itUnit Circle and Trigonometric Ratios — practice 2.1: For an angle in quadrant III, is cosine positive or negative? Answer 1 for positive, −1 for negative.Show answer

Step 1: Locate quadrant III on the coordinate plane. Step 2: Cosine is the x-coordinate on the unit circle. Step 3: Its sign is -1.

Right-Triangle Ratios

Objective: Explain right-triangle ratios and use it to solve or interpret a problem.

For acute angles, sine is opposite/hypotenuse, cosine adjacent/hypotenuse, and tangent opposite/adjacent. These match unit-circle values when scaled. No, it stays opposite the right angle.

Worked examples

Try itUnit Circle and Trigonometric Ratios — practice 3.1: Find tan(3π/4).Show answer

Step 1: Locate 3π/4 on the unit circle. Step 2: Divide its sine by its nonzero cosine. Step 3: The tangent value is -1.

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