Grade 10 · Relate an acute angle to side ratios · 711 of 750
Relate an acute angle to side ratios · practice 73
A right triangle has hypotenuse 11 and a selected acute angle of 65°. Find the opposite and adjacent legs. Follow the calculation, then test the quantities in the animated example.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A right triangle has hypotenuse 11 and a selected acute angle of 65°. Find the opposite and adjacent legs.
Use this scratch space or work on paper. Your notes stay on this page and clear when you leave. Answers are for self-checking; they are not automatically graded.
A starting point
Sine uses opposite/hypotenuse; cosine uses adjacent/hypotenuse for the same angle.
Work through the reasoning
Step 1
Translate the givens
The hypotenuse is 11. Relative to the chosen 65° angle, the desired ratios are opposite/11 and adjacent/11.
Step 2
Calculate with the model
Opposite = 11 sin(65°) ≈ 9.96939; adjacent = 11 cos(65°) ≈ 4.6488.
Step 3
Check and interpret
Using unrounded values, opposite² + adjacent² = 11²(sin²(65°) + cos²(65°)) = 121, confirming the hypotenuse.
The answer
Opposite = 11 sin(65°) ≈ 9.96939; adjacent = 11 cos(65°) ≈ 4.6488.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
For a fixed acute angle, the ratios of corresponding sides are unchanged by scaling a right triangle. Sine compares the opposite leg with the hypotenuse, while cosine compares the adjacent leg with the hypotenuse. The names depend on which acute angle is chosen. Starting quantities: Hypotenuse H = 11; Final acute angle = 65.
opposite = H sin θ; adjacent = H cos θ
03 · Reflect and transfer
Explain what changes and why.
Which leg changes its name if you choose the other acute angle?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.