Angles and sides in trigonometry: Cosine

Now that you've learnt about the triangular function Sine, we'll learn about another trigonometric functions, cosine, and its values. Cosine is also written as cos.

Look at the triangle above. Do you notice the following?

* The unknown angle, α

* The side adjacent to the unknown angle (6.37 in)

* The hypotenuse to the unknown angle (15.3 in)

We need the value of angle, α. Notice that the sides of the triangle are labeled appropriately 'adjacent
side' and hypotenuse' relative to the unknown angle α. Take note that the hypotenuse is not considered
opposite or adjacent to the angle α.

To simplify our discussion, we will simply call the 'length of the adjacent side'simply the 'adjacent'.

Likewise, the other two sides will be called 'opposite' and 'hypotenuse.'

The value for the cosine of angle α is defined as the value that results when you divide the adjacent side by
the hypotenuse. The formula is written below:

cos(α) = adjacent / hypotenuse

Or simply:

cos(α) = adj / hyp

From the diagram above, we can easily solve the unknown angle with the cosine formula:

cos(α) = 6.37 inches / 15.3 in

cos(α) = 0.42

α = cos-1 0.42

α = 65°

Note that the inverse sign is used above. This value can be found using your calculator.


1) If any two values i.e. the 2 sides or, an angle and a side, are given, the missing angle or side can always be found by simply substituting the correct values in the formula.

2) It always helps to draw the diagram to get an accurate picture of what's being asked.

3) Use the calculator to enter the values of cosine and inverse cosine.

Try these questions below

  1. A crate on the ground is attached to a crane's hook suspended from above by a cable that is connected
    to the top of the crane's boom. The length of the boom is 76.78 m. The distance between the base of
    the crane and the crate is 50.5 m. Find the angle between the crate and the base of the crane's body.

    Answer: 49°


    Cos x = 50.5 / 76.78

    Cos x = 0.66

    X = Cos- 0.66

    X= 49°

  2. A steel ladder measuring 23 m long leans against a building and reaches the ledge of a 3rd storey window.
    If the foot of the ladder is 17 m from the foot of the building, (a) how high is the window ledge
    from the ground? (b) What is the angle created between ladder and the ground?

    Answer: (a) 15.5 m (b) 42°


    1. 232 = 172 + x2

      529 = 289 + x2

      529-289 = x2

      √240 = x2

      x = 15.5 m

    2. Cos x = 17 / 23

      Cos x = 0.74 m

      x = Cos- 0.74

      x = 42°

  3. The road leading to Mr. Sanders place inclines upward at 20 degrees from the horizontal ground. How far is the
    section of the road that rises a vertical distance of 310 feet from Mr. Sander's place?

    Answer: 291.37 ft


    Cos 20° = x / 310 ft

    0.9397 x 310 ft = x

    x = 291.37 ft

  4. Molly is standing 150 yards from 38-yard high hill. (a) What angle does she look at to see the top? (b) What is the
    angle from the top looking down to where Molly is? (c) What is the distance between Molly and the top of the hill?

    Answer: (a) 155 yards, (b) 14° (c) 76°


    1. x2 = 382 + 1502

      x= 155 yards

    2. cos x- = 150 / 155

      cos 0.97- = x

      x = 14°

    3. 180 – 90 – 14 = 76°