Trigonometry » Cosine

The following formula is for the cosine:
cos(angle signA) = adjacent shorter side of angle signAhypotenuse

Check adjacent side and hypotenuse if you need more information about this.


How do you use the cosine?

Use the following plan/steps/method:

1. Draw a sketch if it is not yet given.
2. Write down the rule: cos(angle sign...) = ah.
3. Fill in the data that is given.
4. Calculate the unknown value. If necessary, use  2 = 63.


Example 1: Calculate an angle

Question:
Calculate angle signA, round your answer to
one decimal.

Answer:
2. cos(angle signA) = ah
3. cos(angle signA) = 310
4. angle signA = cos-1(310) ≈ 72.5° 
Example with adjacent 3 and hypotenuse 10

Note to step 4:
- On your calculator, you do: [2nd] or [shift] cos (3 : 10) ≈ 72.542...
- Sometimes arccos has to be used instead of cos-1.


Example 2: Calculate a side

Question:
Given is triangle ABC with angle signB = 20°, C = 90° and BC = 10 m.
Calculate the length of AB, round your answer to one decimal.

Answer:

1. Draw a sketch first.
Triangle with angle 20° and adjacent side 10 m
2. cos(angle signB) = ah
3. cos(20°) = 10?
4. Use 2 = 63
The ? is at the location of the 3.
To get 3, you have to do 6 : 2.
Looking back to step 3, we have to do 10 : cos(20°).
AB = 10 : cos(20°) ≈ 10.6 m

Note to step 4:

You do not have to key in the °-sign on the calculator.
Some calculators do not automatically put a '(' behind cos.
If you want you can just key in 10 : cos 20.

Example 3: Calculate a side

Question:
Given is triangle PQR with angle signP = 24°, angle signQ = 90° and PR = 42.
Calculate the length of QR, round your answer to one decimal.

Answer:

1. Draw a sketch first.
Triangle with angle 24° and hypotenuse 42
2. cos(angle signP) = ah
3. cos(24°) = ?42
4. Use 2 = 63
The ? is at the location of the 6.
To get 6, you have to do 2 × 3.
Looking back to step 3, we have to do cos(24°) × 42.
QR = cos(24°) × 42 ≈ 38.4 m

Note to step 4:

You do not have to key in the °-sign on the calculator.
Some calculators do not automatically put a '(' behind cos.
If you want you can just key in cos 24 × 42.