Warm-up
Three quick ones from earlier work. Try each in your head or on paper before you open the answer.
- Two triangles are similar with scale factor 3. A side of the small one is 4 cm. How long is the matching side of the large one?
Answer
12 cm. - Which side of a right-angled triangle is the hypotenuse?
Answer
The side opposite the right angle, and the longest side. - Solve 5x = 20.
Answer
x = 4.
Words used in this lesson
- hypotenuse
- the side opposite the right angle; the longest side. For example: the ramp itself.
- opposite
- the side across from the angle you are using. For example: the height of the ramp.
- adjacent
- the side next to the angle you are using, not the hypotenuse. For example: the ground distance.
- sine
- opposite divided by hypotenuse. For example: sin 30° = 0.5.
- cosine
- adjacent divided by hypotenuse. For example: cos 60° = 0.5.
- tangent
- opposite divided by adjacent. For example: tan 45° = 1.
- angle of elevation
- the angle measured up from the horizontal to what you are looking at. For example: the top of the tree in question 6.
Is a longer ramp a steeper ramp?
Two ramps both make an angle of 20° with the ground. One is 3 m long and one is 6 m long. Does the height of a ramp divided by its length depend on how long the ramp is? Decide before you read on, and say why.
Name the sides from the angle θ: the hypotenuse is opposite the right angle, the opposite side is across from θ, and the adjacent side is the one next to θ that is not the hypotenuse.
sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse, tan θ = opposite ÷ adjacent.
Set an angle and change the hypotenuse. Say which numbers will move and which will not.
θ = 30°, hypotenuse 8: sin θ = 0.5, cos θ = 0.866, tan θ = 0.577.
Show me the steps
- θ = 30°, hypotenuse 8: opposite 4, adjacent 6.93. sin θ = 0.5, cos θ = 0.866, tan θ = 0.577.
- θ = 30°, hypotenuse 12: opposite 6, adjacent 10.39. The ratios are exactly the same as before.
- θ = 60°, hypotenuse 8: opposite 6.93, adjacent 4. sin θ = 0.866 and cos θ = 0.5: the two have swapped.
- θ = 45°, hypotenuse 10: opposite 7.07, adjacent 7.07. They are equal, so tan θ = 1.
Right-angled triangles with the same angle θ are similar (Lesson 5), so their sides are in the same ratios. That is why sin θ, cos θ and tan θ depend only on θ, not on the size of the triangle.
Two slips to watch for. Opposite and adjacent depend on which angle you are using: the opposite side for one acute angle is the adjacent side for the other. And the calculator must be in degree mode, or sin 30° will not be 0.5.
Worked examples
The ramps: 20°, lengths 3 m and 6 m. How high does each rise?
- The height is the side opposite the angle and the ramp is the hypotenuse, so use sin: height = length × sin 20°.
- 3 m ramp: 3 × 0.342 = 1.03 m. 6 m ramp: 6 × 0.342 = 2.05 m.
- Height ÷ length is 0.342 for both. The ratio depends on the angle only, so neither ramp is steeper.
A ladder reaches 4 m up a wall and makes 70° with the ground. How long is the ladder, to 1 decimal place?
- 4 m is opposite the 70° angle and the ladder is the hypotenuse, so sin 70° = 4 ÷ length.
- Rearrange: length = 4 ÷ sin 70° = 4 ÷ 0.9397 = 4.3 m.
Your go. The first step is given. A right-angled triangle has an angle of 25°, with the adjacent side 10 cm. Find the opposite side to 1 decimal place.
- The sides involved are opposite and adjacent, so the ratio to use is tan.
- Write tan 25° = opposite ÷ 10, rearrange and work it out. Then check.
Check the answer
Opposite = 10 × tan 25° = 10 × 0.4663 = 4.7 cm to 1 decimal place.
With a calculator
Check that the calculator is in degree mode: the display shows D or DEG. Press the key sin, cos or tan, type the angle, close the bracket if one appears, then press equals. For sin 65° you should see 0.906307…, and in radian mode you would see a different number. Keep the full calculator value until the last step, then round once.
Check before you go on
1. Triangle ABC has its right angle at C. Which side is opposite the angle at A?
Your turn
Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.
Level 1 · Getting started
2. A right-angled triangle has sides 6, 8 and 10. The angle θ is opposite the side of length 6. Find sin θ.
3. In a right-angled triangle the side opposite θ is 5 and the side adjacent to θ is 12 (the hypotenuse is 13). Find tan θ to 3 decimal places.
Level 2 · Building
4. A ramp 12 m long makes an angle of 35° with the ground. How high does it rise? Give the answer to 1 decimal place.
5. A right-angled triangle has a hypotenuse of 15 cm and an angle of 52°. Find the side adjacent to the 52° angle, to 1 decimal place.
6. You stand 20 m from the foot of a tree. The angle of elevation of its top (the angle up from the ground to the top) is 38°. Ignoring your height, how tall is the tree, to 1 decimal place?
Level 3 · Stretch
7. In a right-angled triangle the side opposite a 28° angle is 7 cm. How long is the hypotenuse, to 1 decimal place?
8. A right-angled triangle has a hypotenuse of 10 cm and an angle of 25°. Find its perimeter, to 1 decimal place.
Something to think about
In every right-angled triangle, is sin θ ever greater than 1? Is tan θ ever greater than 1? Use the figure to look for the biggest and smallest values you can find.
One way to think about it
sin θ and cos θ are always less than 1, because the hypotenuse is the longest side and the opposite and adjacent sides are shorter. tan θ can be more than 1 (when the opposite side is longer than the adjacent side, as at 60 degrees, where tan is 1.732) and it grows without limit as the angle approaches 90 degrees.
Exit ticket
9. A triangle has an adjacent side of 9 and an angle of 40°. Which expression gives the hypotenuse?