Right Triangle Trigonometry — Challenge
One angle and one side are enough to find every other part of a right triangle.
What these problems cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Ratio — A comparison of two quantities, like 3 cups flour to 2 cups water.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- Base — The number being multiplied. In 53 the base is 5.
- Hypotenuse — The side opposite the right angle — always the longest.
- Simplest radical form — No perfect square factors left under the root.
- Adjacent — The side next to that angle which is not the hypotenuse.
- Sine — sin θ = oppositehypotenuse.
- Tangent — tan θ = oppositeadjacent.
- Angle of elevation — The angle up from horizontal to a line of sight.
- Height — The perpendicular distance from the base to the opposite corner.
Problem 1
In a right triangle the side opposite an angle is 3 and the hypotenuse is 5. What is the sine of that angle?
- 35
- 45
- 34
- 53
- 43
Worked answer
35
Sine is the opposite side divided by the hypotenuse.
35, which is 0.6.
Note that a sine can never exceed 1, which rules out 53 immediately.
Problem 2
In a right triangle the side adjacent to an angle is 8 and the hypotenuse is 17. What is the cosine of that angle?
- 1517
- 817
- 815
- 178
- 158
Worked answer
817
Cosine is the adjacent side divided by the hypotenuse.
817.
The third side is 15, since 82+152 = 172, and 1517 would be the sine instead.
Problem 3
What is tan 45°?
- 0
- 12
- 1
- √2
- √3
Worked answer
1
A right triangle with a 45 degree angle has its two legs equal.
Tangent is the opposite divided by the adjacent, and those are the same length here.
So the ratio is 1.
Problem 4
A right triangle has legs 3 and 4 and hypotenuse 5. What is the tangent of the angle opposite the side of length 3?
- 43
- 34
- 35
- 45
- 54
Worked answer
34
Tangent is the opposite side divided by the adjacent side, and the hypotenuse plays no part.
The opposite is 3 and the adjacent is 4, so the tangent is 34.
Taking the other angle would give 43.
Problem 5
What is sin 30°?
- √32
- √22
- 12
- 1
- 13
Worked answer
12
In a 30-60-90 triangle the side opposite the 30 degree angle is half the hypotenuse.
So the sine is 12 exactly.
The value √32 is sin 60°, the other acute angle in the same triangle.
Problem 6
A right triangle has legs 5 and 12. What is the sine of the angle opposite the side of length 5?
- 1213
- 513
- 512
- 135
- 125
Worked answer
513
First find the hypotenuse: √52+122 = √169 = 13.
Sine is opposite over hypotenuse, so it is 513.
512 would be the tangent, which uses the other leg rather than the hypotenuse.
Problem 7
What is cos 60°?
- √32
- 12
- √22
- 1
- 0
Worked answer
12
In a 30-60-90 triangle the side adjacent to the 60 degree angle is half the hypotenuse.
So the cosine is 12.
This equals sin 30°, because the two angles add to 90 degrees.
Problem 8
In a right triangle, what is the side opposite the right angle called?
- The hypotenuse
- The adjacent side
- The opposite side
- The base
- The altitude
Worked answer
The hypotenuse
The side facing the right angle is the hypotenuse.
It is always the longest side, because it faces the largest angle.
Which side counts as adjacent or opposite depends on which acute angle is being considered, but the hypotenuse never changes.