Label the sides relative to the angle you are using, 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.
Elevation and depression
An angle of elevation is measured up from the horizontal; depression is measured down from it.
They are equal
The angle of elevation from A to B equals the angle of depression from B to A. They are alternate angles.
A sanity check
The hypotenuse is always the longest side. If your answer says otherwise, something went wrong.
Choose the ratio that contains what you have
List the sides and angles you know and the one you want, then pick the ratio containing all three. Sine for opposite and hypotenuse, cosine for adjacent and hypotenuse, tangent for opposite and adjacent. The choice is made before any arithmetic.
Label relative to the angle you chose
Opposite and adjacent swap when you switch which acute angle you are working from. Only the hypotenuse keeps its identity. Marking the angle on the diagram before labelling the sides prevents most errors in this topic.
Two sides given means an inverse function
If you know two sides and want the angle, apply sin⁻¹, cos⁻¹ or tan⁻¹ to the ratio. The notation is unfortunate: sin⁻¹(x) is the inverse function, not 1/sin(x), which is the reciprocal and a different thing entirely.
Sanity-check against the triangle
The hypotenuse must be the longest side, and the largest angle sits opposite the longest side. A computed side longer than the hypotenuse, or an angle sum other than 180°, means an error occurred before the arithmetic.
Step 2: Try It Yourself
Tap and try it out.
- sin — opposite over hypotenuse0.60
- cos — adjacent over hypotenuse0.80
- tan — opposite over adjacent0.75
- The marked angle36.87°
The hypotenuse is not given. It comes from 4² + 3² = 25, whose square root is 5.
Step 3: Watch an Example
One step at a time.
Watch Nadia 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 Ratio
Problem 2 of 2
Opposite 5, adjacent 12. What is the tangent, as a decimal to two places?
Solve the Triangle
1 of 4
Legs 7 and 24. Hypotenuse?
2 of 4
Opposite 8, hypotenuse 10. Sine, as a decimal?
3 of 4
Adjacent 6, hypotenuse 10. Cosine, as a decimal?
4 of 4
Which side is always the longest?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Legs 20 and 21. Hypotenuse?
Question 2 of 2
Which ratio involves the opposite and adjacent sides?
What You Learned
- Label the sides relative to your chosen angle.
- Pick the ratio containing what you know and what you want.
- The hypotenuse is always the longest side.