The idea, step by step
Ratios are tied to the reference angle
Sine compares opposite with hypotenuse, cosine compares adjacent with hypotenuse, and tangent compares opposite with adjacent. Similar right triangles explain why a ratio depends on the angle rather than the size of the triangle.
Write a ratio equation containing the unknown and the given side. A ratio with two unknown lengths cannot yet be solved. Check that your calculator is in degree mode for this chapter.
Recover an angle from a ratio
To recover an acute angle, apply the inverse function to a known side ratio. For example, arctangent recovers the angle whose tangent equals the ratio. Inverse sine is not the reciprocal of sine.
Keep calculator precision through the calculation, then round as requested. A hypotenuse must still be the longest side, and all acute angles must lie between 0° and 90°.
A common trap
Writing sin⁻¹(x)=1/sin(x).
Instead: The inverse sine finds an angle. The reciprocal 1/sin(x) is a different operation.
See the thinking
Worked example 1
A right triangle has hypotenuse 12 m and an acute angle of 35°. Find the opposite side. Round to one decimal place.
- Use sine because opposite and hypotenuse are involved.
- sin35°=x/12, so x=12sin35°.
- x=6.8829 m, giving 6.9 m.
Worked example 2
A right triangle has opposite leg 7 cm and adjacent leg 11 cm relative to θ. Find θ to one decimal place.
- Use tanθ=7/11.
- Apply inverse tangent: θ=tan⁻¹(7/11).
- θ=32.4712°, so θ≈32.5°.
Check your understanding
A right triangle has opposite leg 3 and hypotenuse 5. Find sinθ.
Show answer
0.6
sinθ=3/5=0.6.
Your turn
6 questions · 13 marks · Work at your own pace.
Read the questions online
- A right triangle has opposite leg 8 and hypotenuse 17. Find sinθ as a fraction.2 marks
- A right triangle has adjacent leg 12 and hypotenuse 13. Find cosθ.2 marks
- A right triangle has opposite leg 9 and adjacent leg 4. Find tanθ.2 marks
- Find the opposite leg when the adjacent leg is 14 m and θ=28°. Round to one decimal place.2 marks
- Find an acute angle with opposite leg 5 cm and hypotenuse 9 cm. Round to one decimal place.2 marks
- A calculation gives sinθ=1.2 for a right triangle. Can an acute angle satisfy this? Explain.3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A right triangle has opposite leg 8 and hypotenuse 17. Find sinθ as a fraction.2 marks
Sine is opposite divided by hypotenuse.
8/17 - A right triangle has adjacent leg 12 and hypotenuse 13. Find cosθ.2 marks
Cosine is adjacent divided by hypotenuse.
12/13 - A right triangle has opposite leg 9 and adjacent leg 4. Find tanθ.2 marks
Tangent is opposite divided by adjacent.
9/4 or 2.25 - Find the opposite leg when the adjacent leg is 14 m and θ=28°. Round to one decimal place.2 marks
tan28°=x/14.
x=14tan28°.
x≈7.4 m.
7.4 m - Find an acute angle with opposite leg 5 cm and hypotenuse 9 cm. Round to one decimal place.2 marks
sinθ=5/9.
θ=sin⁻¹(5/9).
θ≈33.7°.
33.7° - A calculation gives sinθ=1.2 for a right triangle. Can an acute angle satisfy this? Explain.3 marks
The opposite side is shorter than the hypotenuse, so 0<sinθ<1.
A value 1.2 indicates an incorrect ratio or inconsistent data.
No.
Curriculum reference & lesson scope
SACE Board of South Australia
Stage 1 General Mathematics: practical trigonometry, including similarity, right triangles, triangle area, sine/cosine rules and contextual interpretation.
Official curriculum source ↗