Trigonometric Ratios (I)
Geometry · Trigonometric Ratios (I)
Syllabus tag: KCSE | Mathematics | Form 2 | Topic 9 Trigonometric Ratios (I)
Lesson objectives
By the end of this topic, you should be able to:
- Define tangent, sine and cosine ratios from a right-angled triangle.
- Read and use tables of trigonometric ratios.
- Use the ratios to solve for unknown sides and angles.
- Apply trigonometry to problems of elevation, depression and bearings.
Trigonometric Ratios (I)
Trigonometry connects the angles of a right-angled triangle with the ratios of its sides.
a) Naming the sides
The hypotenuse is opposite the right angle. It never changes.
The opposite side is across from the angle you are working with.
The adjacent side is next to that angle, and is not the hypotenuse.
Opposite and adjacent swap over if you change which angle you are using. The hypotenuse does not.
b) The three ratios
Remember them as SOH CAH TOA.
These ratios depend only on the angle, not on the size of the triangle. A 30° angle gives sin 30° = 0.5 in any right-angled triangle, large or small.
That is precisely why the tables work.
c) Choosing the right ratio
Label the sides relative to the angle you are given or want.
Note which two sides are involved: the one you know and the one you want.
Choose the ratio containing exactly those two.
If it involves the hypotenuse and the opposite, use sine. Hypotenuse and adjacent, cosine. Opposite and adjacent, tangent.
d) Finding a side
Substitute into the chosen ratio, then rearrange.
If the unknown is on top, multiply. If it is underneath, divide.
e) Finding an angle
If two sides are known, form the ratio and work backwards.
For opposite 7 and hypotenuse 14, sin θ = 0.5, so θ = 30°.
Use the inverse function, written sin⁻¹, or read the tables in reverse.
f) Special angles worth knowing
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 0.5 | 0.866 | 0.577 |
| 45° | 0.707 | 0.707 | 1 |
| 60° | 0.866 | 0.5 | 1.732 |
| 90° | 1 | 0 | undefined |
Notice sin and cos swap between 30° and 60°. Notice tan 45° = 1, since the two shorter sides are equal there.
g) Elevation, depression and bearings
The angle of elevation is measured upwards from the horizontal.
The angle of depression is measured downwards from the horizontal.
They are equal between two points, because the horizontals are parallel and these are alternate angles.
For bearings, draw the north line, mark the angle clockwise, and find the right-angled triangle inside the figure.
Always sketch first, mark the right angle, then label the sides. Most errors come from starting to calculate before the sketch is right.
h) Where this is used
Finding heights without climbing. Surveying. Navigation. Any question giving one side and one angle of a right-angled triangle.
Words to know
- Hypotenuse -- the side opposite the right angle.
- Opposite side -- the side across from the angle being used.
- Adjacent side -- the side beside the angle, between it and the right angle.
- Inverse ratio -- the operation finding an angle from its trigonometric ratio.
- Angle of elevation -- the angle above the horizontal when looking up.
:::checkpoint Check yourself
- Write the three ratios in the SOH CAH TOA form.
- A right-angled triangle has hypotenuse 10 cm and an angle of 60°. Find the opposite side.
- If tan θ = 1, what is θ?
- Why do the ratios depend only on the angle, not the triangle's size? :::
Bridge to practice
The exercises begin with naming sides and stating the ratios, move through finding sides and angles and using tables, and finish with elevation, depression and bearings. For every question, mark the angle first, then label opposite and adjacent relative to it.