Find an acute angle
Use the two given lengths to form a ratio for the requested angle. Apply the matching inverse in degree mode
- Choose from the two given sides Name the sides from the requested angle, then use the ratio containing both lengths \(\displaystyle \begin{aligned}\sin\theta&=\frac{\text{opposite}}{\text{hypotenuse}}\\\cos\theta&=\frac{\text{adjacent}}{\text{hypotenuse}}\\\tan\theta&=\frac{\text{opposite}}{\text{adjacent}}\end{aligned}\)
- Use the matching inverse Enter the fraction directly with the matching inverse key in degree mode. Keep the calculator value until the final rounding
- Check the angle The result must be acute. A tangent ratio greater than one is valid and gives the larger acute angle. The two acute angles together make a right angle \(\displaystyle \alpha+\beta=90^{\circ}\)
Worked example
The opposite length and hypotenuse of a right triangle are given for the acute angle. Find the angle to one decimal place in degree mode
\[\begin{aligned}o&=18.7\text{ cm}\\h&=31.4\text{ cm}\end{aligned}\]
- \(\displaystyle \sin\theta=\frac{18.7}{31.4}\)Sine uses the opposite side and hypotenuse
- \(\displaystyle \theta=\sin^{-1}\left(\frac{18.7}{31.4}\right)\)Use the inverse key in degree mode. Enter the original fraction directly
- \(\displaystyle \theta=36.551236\ldots^{\circ}\)Keep the calculator result before rounding
- \(\displaystyle \theta\approx36.6^{\circ}\)Round once to one decimal place. The result is acute
Common mistakes
Using a different inverse function
✕\(\displaystyle \begin{aligned}\sin\theta&=0.718\\\theta&=\cos^{-1}(0.718)\end{aligned}\)✓\(\displaystyle \begin{aligned}\sin\theta&=0.718\\\theta&=\sin^{-1}(0.718)\end{aligned}\)
Match the inverse key to the ratio you formed. Changing sine to cosine changes the angle being recoveredUsing the quotient as an angle
✕\(\displaystyle \begin{aligned}\tan\theta&=1.73\\\theta&=1.73^{\circ}\end{aligned}\)✓\(\displaystyle \begin{aligned}\tan\theta&=1.73\\\theta&=\tan^{-1}(1.73)\end{aligned}\)
A side quotient has no angle unit. Apply the inverse function to recover the angleDifficulty levels
introFind an angle from opposite and hypotenuse lengths, to the nearest degree
standardUse sine, cosine or tangent to find an angle to one decimal place
examChoose an angle from a supplied right triangle
Questions about this topic
Should I round the side quotient first?+
No. Enter the original fraction directly or keep the full calculator value. Round only the requested angle
Can a tangent ratio exceed one?+
Yes. It means the opposite leg is longer than the adjacent leg. The angle is still acute
How can I find the other acute angle?+
Subtract the unrounded first angle from a right angle and round the requested result. You can also swap the side roles and calculate the other angle directly
Can I use a calculator?+
A calculator is allowed unless a question is marked no calculator
Which year level and chapter is this?+
Year 10 Mathematics, chapter 9: Pythagoras and trigonometry
In this chapter
9A Finding the hypotenuse: Find a side of a right-angled triangleDetermine whether three sides form a right-angled triangle9B Finding a shorter side: Find a side of a right-angled triangle9C Pythagoras' theorem in 2D and 3D: Find a side of a right-angled trianglePythagoras in three dimensionsName sides and state trigonometric ratios9D Trigonometric ratios: Evaluate a trigonometric ratio9D Trigonometric ratios: Choose a trigonometric ratio9E Unknown side in the numerator: Find an unknown side of a right-angled triangle9E Unknown side in the numerator: Solve a practical right-angled-triangle problem9E Unknown side in the numerator: Evaluate a trigonometric ratio9E Unknown side in the numerator: Choose a trigonometric ratio9F Unknown side in the denominator: Find an unknown side of a right-angled triangle9F Unknown side in the denominator: Solve a practical right-angled-triangle problem9F Unknown side in the denominator: Choose a trigonometric ratioFind an angle of a right-angled triangle from two sides9G Finding angles with trigonometry: Solve a practical right-angled-triangle problem9G Finding angles with trigonometry: Evaluate a trigonometric ratio, and find an angle from its valueAngles of elevation and depressionRead and state three-figure bearingsSolve a navigation problem with bearingsTrigonometry in three dimensions