Label the sides relative to your angle, then pick the ratio containing what you know and what you want.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Finding a side
Set up the ratio, substitute, and solve. If the unknown is on the bottom, multiply up before dividing.
Finding an angle
When two sides are known, use an inverse: arcsin, arccos or arctan gives the angle back.
Elevation and depression
An angle of elevation is measured up from the horizontal; depression is measured down from it. The two are equal between the same pair of points.
Check the answer
The hypotenuse must be longest and any acute angle below 90°. A violated check means a wrong ratio.
Do not forget Pythagoras
With two sides known, Pythagoras finds the third faster than any trigonometry.
Choose the ratio that has what you know
List what is given and what is wanted, then pick the ratio containing all three quantities. Making that choice before any arithmetic keeps these problems mechanical.
Finding an angle uses the inverse
Two known sides give a ratio; the inverse function converts it back to an angle. sin⁻¹ is the inverse function, not the reciprocal, despite what the notation suggests.
Check the calculator mode
Degrees or radians. A wrong mode gives confidently wrong answers with no warning. Verifying that sin(30) returns 0.5 takes two seconds and settles it.
Check against the picture
The hypotenuse must be longest, and the larger angle sits opposite the longer side. A computed side exceeding the hypotenuse signals an error before the arithmetic is even reviewed.
Step 2: Try It Yourself
Tap and try it out.
- sin — opposite over hypotenuse0.71
- cos — adjacent over hypotenuse0.71
- tan — opposite over adjacent1
- The marked angle45°
The hypotenuse is not given. It comes from 5² + 5² = 50, whose square root is 7.07.
Step 3: Watch an Example
One step at a time.
Watch Marcus Find a Height
From 50 m away, the angle of elevation to a tower top is 30°.
- Step 1
The 50 m is adjacent to the angle and the height is opposite it.
Step 4: Your Turn
Practice makes it stick.
The Ladder
Problem 1 of 2
A right triangle has legs 9 and 12. What is the hypotenuse?
The Equal Legs
Problem 2 of 2
A right triangle has equal legs. What is each acute angle, in degrees?
degrees
Solve the Triangle
1 of 8
Legs 7 and 24. Hypotenuse?
2 of 8
Legs 8 and 15. Hypotenuse?
3 of 8
tan θ = 1. What is θ, in degrees?
4 of 8
sin θ = 0.5. What is θ, in degrees?
5 of 8
Which ratio holds opposite and adjacent? 1 sine, 2 cosine, 3 tangent.
6 of 8
Hypotenuse 20, opposite 10. Sine as a decimal?
7 of 8
Put the solving process in order.
- 1Label the sides opposite, adjacent and hypotenuse.
- 2Choose the ratio holding what you know and want.
- 3Substitute and solve, then check the answer is sensible.
- 4Mark the angle you are working from.
8 of 8
Legs 20 and 21. Hypotenuse?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Legs 6 and 8. What is the hypotenuse?
Question 2 of 2
Two sides are known and you want the angle. What do you use?
What You Learned
- Choose the ratio containing what you know and what you want.
- Use an inverse function when the unknown is an angle.
- Check that the hypotenuse is longest and the angles are acute.