Sin, cos and tan

Before we can use trigonometric relationships we need to understand how to correctly label a right-angled triangle.

Right-angled triangle with hypotenuse, opposite side, adjacent side, and an internal angle labelled accordingly

There are three labels we will use:

  • Hypotenuse - The longest side of a triangle. This will always be opposite the right angle.
  • Opposite - This is the side opposite the angle you are using.
  • Adjacent - This is the remaining side. It should join to the hypotenuse to form the angle we are using.

Let’s look at how to use trigonometric identities to calculate missing sides.

{sin~θ} = \frac {opposite} {hypotenuse}

{cos~θ} = \frac {adjacent} {hypotenuse}

{tan~θ} = \frac {opposite} {adjacent}

Adjacent, opposite and hypotenuse signify the length of these sides respectively.

\text{θ} signifies the size of the angle in the triangle.

A useful way to remember these three equations is the acronym SOHCAHTOA which stands for:

SOH

{sin}~=~\frac{opposite}{hypotenuse}

CAH

{cos}~=~\frac{adjacent}{hypotenuse}

TOA

{tan}~=~\frac{opposite}{adjacent}

If we know one of the two angles inside the triangle (not including the right angle) and the length of any of the three sides, we can calculate all the other measurements for the shape.

Example 1

What is the length of the side marked {x}?

Right-angled triangle where the hypotenuse equals 15 cm, the angle equals 53 degrees, and the opposite side is labelled x

Firstly we need to work out what we know.

We know that the hypotenuse is of length 15 cm and that the angle θ is 53°.

We need to calculate the opposite side. This means that we must use the equation for sin:

{sin~θ} = \frac {opposite} {hypotenuse}

Rearranging we get:

{hypotenuse} \times~{sin~θ}~=~{opposite}

Substituting in the values we know, gives:

{15} \times~{sin}~{53\circ}~=~{opposite}

We get this value by pressing 15, then the × button, then the sin button on the calculator, followed by 53 (Note that on new calculators we don’t need to press the × button).

This gives us: opposite = 11.97953265 cm

So the opposite side has a length of 12 cm (to the nearest cm).

A quick check when calculating the adjacent and opposite sides is to make sure that your answer is less than the length of the hypotenuse.

Question

Find the length of the side BC.

Right-angled triangle ABC where the hypotenuse AB equals 7 cm, and the angle equals 30 degrees

Side BC is opposite the 30° angle. We also know the hypotenuse is 7 cm in length. We must use the equation for sin:

{sin~θ} = \frac {opposite} {hypotenuse}

Rearranging we get:

{hypotenuse} \times~{sin~θ}~=~{opposite}

Substituting in the values we know, gives:

{7} \times~{sin~30\circ}~=~{opposite}

We obtain the value of sin by using the sin button on the calculator, followed by 30.

This gives us: opposite = 3.5 cm.

So the length of BC is 3.5 cm.

Example 2

Find the length of the side YZ.

Right-angled triangle XYZ where the angle equals 25 degrees, and the adjacent side XZ equals 5 cm

This time we know the adjacent side and we want to find out the hypotenuse. We therefore select the equation for cos as it contains both these terms.

{cos~θ} = \frac {adjacent} {hypotenuse}

Rearranging we get:

{hypotenuse} = \frac {adjacent} {cos~θ}

What we have done here is manipulated the equation by multiplying both sides by the hypotenuse and dividing both sides by cos θ.

This effectively swaps cos θ with the denominator of the fraction, the hypotenuse. This will always work for any of the three equations.

Substituting in the values we know gives:

{hypotenuse} = \frac {5} {cos~25\circ}

We obtain the value of cos 25° by using the cos button on the calculator, followed by 25.

This gives us: hypotenuse = 5.516889595 cm.

So the length of YZ is 5.52 cm (to two decimal places).