Maths

Trigonometry - Higher

Trigonometry can be used to calculate the lengths of sides and sizes of angles in **right-angled triangles**.

The sides of the right-angled triangles are given special
names - the **hypotenuse**, the **opposite** and the **adjacent**.

The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides relate to the angle under consideration.

Click on the correct side of the triangle on the diagram below.

There are three formulae involved in trigonometry:

sin = ^{opposite} / _{hypotenuse}

cos = ^{adjacent} / _{hypotenuse}

tan = ^{opposite} / _{adjacent}

Which formula you use will depend on the information given in the question.

There are a couple of ways to help you remember which formula to use. Remember SOHCAHTOA (it sounds like 'Sockatoa') or **S**ome **O**ld **H**ag **C**racked **A**ll **H**er **T**eeth **O**n **A**pples.

- Question
Find the length of side BC.

- Answer
- We are given angle
**A**and side**AB**. **AB**is the**hypotenuse**.**BC**is**opposite**angle A.- Therefore we use the formula:
- sin =
^{opposite}/_{hypotenuse} - sin 30 =
^{BC}/_{7} - Multiply both sides by 7
- 7 x sin30 = BC
- Using a calculator we get
**BC = 3.5cm**

- We are given angle

Now you try this one.

- Question
Find length of side PQ, giving your answer correct to 3 sf.

- Answer
The answer is

**4.77cm**(3 sf).- Can you see that PQ is
**opposite**the given angle and that QR is**adjacent**to it? - Use the formula:
- tan =
^{opposite}/_{adjacent} - tan50 =
^{PQ}/_{4} - 4 × tan50 = PQ
**PQ = 4.77**(3 sf)

- Can you see that PQ is

- Question
Find YZ, giving your answer correct to 3 sf

- Answer
- We are given angle
**Z** **XZ**is**adjacent**to angle Z.**YZ**is the**hypotenuse**.- Therefore we use the formula: cos =
^{adjacent}/_{hypotenuse} - This time our unknown side (YZ) is the denominator. If we multiply both sides by YZ, we get:
**YZ x cos25 = 5**- Now we can divide both sides by cos25:
- Using a calculator:
- YZ = 5 ÷ 0.9063 = 5.52cm (3 sf)

- We are given angle

Look again at the original arrangement of this equation. By rearranging it we have swapped the positions of YZ and cos25. This method works whenever the unknown side is at the bottom of the fraction.

Now have a go at this question, using the above method.

- Question
Find PQ, giving your answer correct to 3 sf

- Answer
The answer is

**7.15cm**.We have been given angle Q. Side PR is opposite angle Q and PQ is adjacent to it. We have the opposite and the adjacent so we use the formula:

tan = opposite / adjacent

- tan40 = 6 / PQ
- By rearranging the formula we get
- PQ = 6 / tan40
- Using a calculator
- PQ = 7.15 (3 sf)

- Question
Find angle C. Give your answer correct to 1 dp.

- Answer
We have been given the lengths of AC and BC and asked to find angle

**C**.**AC**is**adjacent**to angle C.**BC**is the**hypotenuse**.- So we use the formula:
- cos =
^{adjacent}/_{hypotenuse} - cosC =
^{5}/_{7} To calculate C we need to find the

**inverse**of cos (INV cos or SHIFT cos):C =

**Inv cos**(^{5}/_{7})Using a calculator:

**C = 44.4°**

Make sure that you know how your calculator works.

**Use your calculator to find Inv cos**(^{5}/_{7}), and check that you get the right answer (44.4°).

If you forget how to do it use this as a test:

The inverse cos of 1 is zero. **Inv cos 1 = 0** (because cos0 = 1)

- Question
Find angle Z, giving your answer correct to 1 dp.

- Answer
Z =

**38.7°**You have been given the

**opposite**and the**hypotenuse**. Therefore:sin Z =

^{5}/_{8}Z = inv sin(

^{5}/_{8})**Z = 38.7°**(1 dp)

Questions involving trigonometry and right-angled triangles will only require you to know these three formula (sin, cos and tan). Make sure that you are confident with them.

Practise recognising the sides of a triangle, and calculating their lengths in this activity.

Now try a Test Bite