To find a missing angle, name the two sides you know, write the ratio as a fraction, then press the inverse function on your calculator while it is in degree mode. The result is the angle in degrees, usually rounded to 1 decimal place.
This lesson follows choosing sine, cosine or tangent and uses the same labelling, because the ratio still comes from the sides.
What does the inverse button actually do?
Sine takes an angle and gives a ratio: sin 30° = 0.5. Sin⁻¹ goes the other way: sin⁻¹ 0.5 = 30°.
The small −1 does not mean “one over”. It means “undo sine”.
So when a question gives two sides and asks for an angle, you build a ratio from the sides first. The inverse then turns that ratio into the angle.
How do you find the angle, step by step?
- Label the sides from the angle you want: hypotenuse, opposite, adjacent.
- Pick the ratio from the two sides you know.
- Write the equation with the angle as θ, for example sin θ = 7/10.
- Apply the inverse: θ = sin⁻¹(7/10).
- Check the mode is degrees, and type the fraction in brackets.
- Round to 1 decimal place unless the question says otherwise.
Worked example
In a right-angled triangle, the side opposite angle θ is 7 cm and the hypotenuse is 10 cm. Find θ.
Step 1, label: opposite = 7, hypotenuse = 10, adjacent not needed.
Step 2, ratio: opposite and hypotenuse means sine.
Step 3, equation: sin θ = 7/10 = 0.7.
Step 4, inverse: θ = sin⁻¹(0.7) = 44.427…°
Step 5, round: θ = 44.4° (1 d.p.).
Check: 44.4° is an acute angle and is less than 90°, as it must be. A sine of 0.7 is between sin 30° = 0.5 and sin 60° = 0.866, so an angle between 30° and 60° is right.
The mistake to watch for
A common slip is to have the calculator in the wrong mode, or to press sin instead of sin⁻¹.
Mistaken working: the student types sin(0.7) in degree mode and writes θ = 0.0122°. Or the student types sin⁻¹(0.7) in radian mode and writes θ = 0.775.
Both answers are tiny and suspicious for an angle in a triangle with a side of 7 and a hypotenuse of 10.
The correction is two checks. First, use the inverse key when the angle is the unknown. Second, test the mode with sin⁻¹(0.5), which must show 30.
Also ask whether the angle looks plausible: 7 is most of 10, so the angle should be fairly large.
Check yourself
Give each angle to 1 decimal place.
1. Opposite = 9 cm, adjacent = 12 cm. Find θ.
Show answer
Opposite and adjacent means tangent: tan θ = 9/12 = 0.75, so θ = tan⁻¹(0.75) = 36.9°.
2. Adjacent = 5 cm, hypotenuse = 13 cm. Find θ.
Show answer
Adjacent and hypotenuse means cosine: cos θ = 5/13, so θ = cos⁻¹(5/13) = 67.4°.
3. Opposite = 4 cm, adjacent = 7 cm. Find θ and say why the answer is sensible.
Show answer
tan θ = 4/7, so θ = tan⁻¹(4/7) = 29.7°. The opposite side is shorter than the adjacent side, so the angle should be below 45°, and 29.7° is.
Where this leads next
Next, combine Pythagoras with trigonometry for questions that need two steps. The right-triangle trigonometry practice set has angle questions mixed in with side questions. The non-calculator working trainer helps with simplifying fractions such as 9/12 before you press any button.
If your angles are right on good days and wrong on others, a settings slip may be hiding. Our teachers look for that kind of habit in online one-to-one Mathematics tuition.