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Grade 10 · lesson 2 of 4

Use Opposite over Hypotenuse

For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length.

About 20 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 4 cm. Find its opposite side. Candidate answers: (A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm.
  • Independent check inputs: In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 6 cm. Find its opposite side. Candidate answers: (A) 2 cm (B) 3 cm (C) 4 cm (D) 6 cm.

Learn the idea

Similarity preserves angle measures and corresponding length ratios. Right triangles with the same acute angle are similar, explaining why sine, cosine and tangent depend on an angle rather than the triangle’s overall size. For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length. Multiplying the hypotenuse by the sine recovers the opposite leg.

Here is the method on the supplied teaching case. Sine is opposite/hypotenuse. If o/4=1/2, then o=4/2=2 cm. The side opposite θ depends on which acute angle was chosen.

The opposite side changes when the reference angle changes. The hypotenuse remains the side opposite the right angle; do not use an adjacent leg in the sine denominator.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 4 cm. Find its opposite side. Candidate answers: (A) 1 cm (B) 2 cm (C) 3 cm (D) 4 cm. Solution: Sine is opposite/hypotenuse. If o/4=1/2, then o=4/2=2 cm. The side opposite θ depends on which acute angle was chosen. The final answer is 2 cm. The opposite side changes when the reference angle changes. The hypotenuse remains the side opposite the right angle; do not use an adjacent leg in the sine denominator.

Your turn

Check your reasoning

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Helpful tips
  • For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length.
  • The opposite side changes when the reference angle changes.
In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 6 cm. Find its opposite side.

Feedback appears here after you check. This is practice, not a graded school assessment.

See the answer and explanation

3 cm

Sine is opposite/hypotenuse. If o/6=1/2, then o=6/2=3 cm. The side opposite θ depends on which acute angle was chosen.

Take it further

Changed-input task: In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 18 cm. Find its opposite side. Candidate answers: (A) 10 cm (B) 18 cm (C) 8 cm (D) 9 cm. Show the mathematical reason, not only the selected answer. Expected result: 9 cm. Sine is opposite/hypotenuse. If o/18=1/2, then o=18/2=9 cm. The side opposite θ depends on which acute angle was chosen.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 8 cm. Find its opposite side.
Need a hint?
  • For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length.
  • The opposite side changes when the reference angle changes.
2. In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 10 cm. Find its opposite side.
Need a hint?
  • For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length.
  • The opposite side changes when the reference angle changes.
3. In a right triangle, an acute angle θ has sin θ=1/2 and hypotenuse 12 cm. Find its opposite side.
Need a hint?
  • For an acute angle in a right triangle, sine is opposite-leg length divided by hypotenuse length.
  • The opposite side changes when the reference angle changes.

Your choices are temporary. Feedback appears after you check all the answers.

Nothing here is saved to an account or sent to a teacher. Keep your own notes or download them before leaving. The score describes these questions; it does not certify subject mastery.

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