Right-Triangle Trigonometry

Develop ratios and similarity, reference angles across quadrants.

Lesson 3 of 32 parts Practice Flashcards

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Ratios and Similarity

Objective: Explain ratios and similarity, apply the rule, and justify relevant restrictions.

For an acute angle θ in a right triangle, sine is opposite/hypotenuse, cosine is adjacent/hypotenuse, and tangent is opposite/adjacent. Similar right triangles preserve these ratios, so the values depend on the angle rather than overall triangle size. Label opposite and adjacent relative to the chosen angle, not the page orientation. The hypotenuse is always opposite the right angle.

Formulas

$sin θ=opp/hyp; cos θ=adj/hyp; tan θ=opp/adj$

Worked examples

Try itA right triangle has legs 21 and 28, and hypotenuse 35. For the acute angle opposite the 21 leg, find sin θ.Show answer

Step 1: Sine equals opposite/hypotenuse. Step 2: Use 21/35. Step 3: Reduce to 3/5; triangle scale does not change the ratio.

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Ratios and Similarity

Objective: Explain ratios and similarity, apply the rule, and justify relevant restrictions.

Formulas

$sin θ=opp/hyp; cos θ=adj/hyp; tan θ=opp/adj$

Worked examples

Try itA right triangle has legs 21 and 28, and hypotenuse 35. For the acute angle opposite the 21 leg, find sin θ.Show answer

Step 1: Sine equals opposite/hypotenuse. Step 2: Use 21/35. Step 3: Reduce to 3/5; triangle scale does not change the ratio.

Reference Angles Across Quadrants

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

Reference angles extend right-triangle magnitudes to angles outside the first quadrant. For θ in quadrant II, the reference angle is π−θ; in quadrant III it is θ−π; in quadrant IV it is 2π−θ, after reduction to [0,2π). Attach signs using unit-circle coordinates. A reference angle alone does not identify the original quadrant or terminal ray.

Formulas

In QII: θref=π−θ

Worked examples

Try itFind the reference angle for 104°.Show answer

Step 1: 104° lies in quadrant II. Step 2: In quadrant II, reference angle = 180°−θ. Step 3: Compute 180°−104°=76°.

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