Angles and sides in trigonometry: Sine

Now that you've learnt about the Pythagorean theorem, we'll focus on the angular relationships of the
triangular sides with each other and the whole triangle as a whole. In real life, this is useful navigation
technique in extrapolating distances.

Particularly, we'll learn about one of the most common the trigonometric functions, sine, and its values. Sine
is also written as sin.

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

* The unknown angle, α

* The side opposite 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 'opposite 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 opposite side'simply the 'opposite.'

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

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

sin(α) = opposite / hypotenuse

Or simply:

sin(α) = opp / hyp

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

sin(α) = 6.37 inches / 15.3 in

sin(α) = 0.42

α = sin-1 0.42

α = 24.6°

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 sine and inverse sine.

Try these questions

  1.   A ladder measuring 10.45 m leans against a building. If the ladder makes an angle of 60° with the
    ground, (a) how far up the building wall does the ladder reach?

    Answer: 9.1 m


    Solve for the opposite side using sine since the hypotenuse and angle is given.

    Sin (60°) = x / 10.45

    Sin (60°) x 10.45 = x

    0.87 x 10.45 = x

    9.1 m = x

  2.   The construction of a ramp for wheelchairs is currently underway in the new building facility in St.
    Luke's Medical Center. The building inspector needs to find the angle of the ramp to see if it meets
    regulations. With a tape measure, he found that the stage is 5t high and the distance along the ramp
    is 35ft.

    Answer: 8°


    Solve for the angle using sine since the hypotenuse and opposite sides are given.

    Sin x° = 5 ft / 35 ft

    Sin x° = 0.14

  3. Sin-1 0.14

    = 8°

  4.   A flat 21 foot ramp rests with one end on the ground and the other end on a 3 foot ledge.

    (a) What is the measure of the angle created here?

    (b) How far from the base of the ledge is the far end of the plank?



    a) First, find the angle:

    Sin x = 3 ft / 21 ft

    Sin x = 0.14

    Sin-1 0.14

    = 8°

    b) Second, use Pythagorean theorem to find the missing side

    212 = 32 + x2

    441 = 9 + x2

    432 = x2

    X = √432

    X = 20.8 ft

  5.   Mike flew his kite one sunny day. It is tied to a string that is 560 ft long. If the string makes an angle 50° with
    Mike on the ground, what is the vertical height of the kite?

    Answer: 1376 m



    Sin 50° = x / 560

    0.770 x 560 = x

    431 ft = x