- Trigonometry is the mathematics of
**angles**in triangles - It looks at the relationship between side lengths and angles of triangles
- It comes from the Greek words
*trigonon*meaning ‘triangle’ and*metron*meaning ‘measure’

- The three trigonometric functions Sine, Cosine and Tangent come from ratios of side lengths in
**right-angled**triangles - To see how the ratios work you must first label the sides of a right-angled triangle in relation to a chosen angle
- The
**hypotenuse,**is the*H*,**longest side**in a right-angled triangle- It will always be
**opposite**the right angle

- It will always be
- If we label one of the other angles θ, the side opposite
**opposite**,and the*O,***adjacent**,*A*

- The
- The functions Sine, Cosine and Tangent are the ratios of the lengths of these sides as follows

**SOHCAHTOA**is a mnemonic that is often used as a way of remembering which ratio is which**S**in is**O**pposite over**H**ypotenuse**C**os is**A**djacent over**H**ypotenuse**T**an is**O**pposite over**A**djacent

- In a right-angled triangle, label one angle other than the right angle and label the sides of the triangles as follows

- Note that
**θ**is the Greek letter theta**O**= opposite θ**A**= adjacent (next to) θ-
**H**= hypotenuse - '**H**' is always the same, but 'O' and 'A' change depending on which angle we're calling θ

- Using those labels, the three
**SOHCAHTOA**equations are:

- If you know the length of one of the sides of any right-angled triangle and one of the angles you can use SOHCAHTOA to find the length of the other sides
- Always start by
**labelling the sides**of the triangle with H, O and A - Choose the correct ratio by looking only at the values that you have and that you want
- For example if you know the angle and the side opposite it (O) and you want to find the hypotenuse (H) you should use the sine ratio

- Substitute the values into the ratio
- Use your calculator to find the solution

- Always start by

** **

- If you know two sides of any right-angled triangle you can use SOHCAHTOA to find the size of one of the angles
- Missing angles are found using the
**inverse functions**:

, ,

- After choosing the correct ratio and substituting the values use the inverse trigonometric functions on your calculator to find the correct answer

- SOHCAHTOA (like Pythagoras) can only be used in right-angles triangles – for triangles that are not right-angled, you will need to use the Sine Rule or the Cosine Rule
- Also, make sure your calculator is set to measure angles in degrees

Find the values of and in the following triangles.

Give your answers to 3 significant figures.

To find , first label the triangle

We know A and we want to know O - that's TOA or

Multiply both sides by 9

Enter on your calculator

Round to 3 significant figures

To find , first label the triangle

We know A and H - that's CAH or

Use inverse cos to find

Enter on your calculator

Round to 3 significant figures

- If a person looks at an
**object**that is not on the same horizontal line as their eye-level they will be looking at either an angle of**elevation**or**depression**- If a person looks
**up**at an object their line of sight will be at an**angle of elevation**with the horizontal - If a person looks
**down**at an object their line of sight will be at an**angle of depression**with the horizontal

- If a person looks
- Angles of elevation and depression are measured
**from the horizontal** **Right-angled trigonometry**can be used to find an angle of elevation or depression or a missing distance- Tan is often used in real-life scenarios with angles of elevation and depression
- For example if we know the distance we are standing from a tree and the angle of elevation of the top of the tree we can use Tan to find its height
- Or if we are looking at a boat at to sea and we know our height above sea level and the angle of depression we can find how far away the boat is

- It may be useful to draw more than one diagram if the triangles that you are interested in overlap one another

A cliff is perpendicular to the sea and the top of the cliff stands 24 m above the level of the sea. The angle of depression from the cliff to a boat at sea is 35°. At a point m up the cliff is a flag marker and the angle of elevation from the boat to the flag marker is 18°.

a)

Draw and label a diagram to show the top of the cliff, T, the foot of the cliff, F, the flag marker, M, and the boat, B, labelling all the angles and distances given above.

b)

Find the distance from the boat to the foot of the cliff.

c)

Find the value of .

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