TRIGONOMETRY

Mathematics can sometimes feel intimidating, especially when you hear the word Trigonometry. Many students think it’s only about complicated formulas and calculators. But here’s the truth: Trigonometry is simply about right-angled triangles, and how their sides and angles are connected.

If you can understand the three special ratios (sine, cosine, and tangent), you will unlock the secrets of solving heights, distances, ladders, ramps, and even real-life navigation problems.

This guide will break Trigonometry into easy-to-follow steps, give you clear worked examples, and finally provide 20 practice questions so you can test yourself and walk confidently into your exams. 

So, let's get started!

We'll take it in this order...

v  Introduction to Trigonometry

v  Trigonometric Ratios (SOH-CAH-TOA)

v  Using Ratios to Find Missing Sides

v  Using Ratios to Find Missing Angles

v  Angles of Elevation and Depression (Word Problems)

v  Worked Examples

v  20 Practice Questions for You

 

1. Introduction to Trigonometry

• Trigonometry is about the relationship between angles and sides of right-angled triangles.

• You must have a right angle (90°) in the triangle to apply basic trigonometry.

We label the sides of the triangle like this:

• Hypotenuse (H): longest side, opposite the right angle.

• Opposite (O): side opposite the angle you are working with.

• Adjacent (A): side next to the angle you are working with, but not the hypotenuse.

2. Trigonometric Ratios (SOH-CAH-TOA)

We use three main ratios:

$$\sin \theta =\frac{\text{Opposite}}{\text{Hypotenuse}}$$

   ​

$$\cos \theta =\frac{\text{Adjacent}}{\text{Hypotenuse}}$$$$\tan \theta =\frac{\text{Opposite}}{\text{Adjacent}}$$

👉 Memory tip: SOH-CAH-TOA

3. Finding Missing Sides

• If you know one angle (other than 90°) and one side, you can find another side using the ratios.

Example:

$$\sin \theta =\frac{O}{H}\Rightarrow O=H\times \sin \theta$$

4. Finding Missing Angles

• If you know two sides, you can find the angle by using inverse trig functions:

$$\theta ={\sin }^{-1}\left(\frac{O}{H}\right),\theta ={\cos }^{-1}\left(\frac{A}{H}\right),\theta ={\tan }^{-1}\left(\frac{O}{A}\right)$$

Use your calculator for this.

5. Angles of Elevation & Depression

• Elevation: looking up at something (angle measured upwards).

• Depression: looking down at something (angle measured downwards).

These are solved using the same SOH-CAH-TOA method, but in word problems.

6. Worked Examples

Example 1: Finding a side

In a right triangle, 

$\theta ={30}^{\circ }$

$$\sin {30}^{\circ }=\frac{O}{10}$$

​

$$0.5=\frac{O}{10}$$$$O=10\times 0.5=5\text{cm}$$

Example 2: Finding an angle

In a right triangle, the opposite = 4 cm and adjacent = 3 cm. Find angle 

$\theta$

$$\tan \theta =\frac{O}{A}=\frac{4}{3}$$

​

$$\theta ={\tan }^{-1}\left(\frac{4}{3}\right)$$

On calculator: 

$\theta \approx {53.1}^{\circ }$

                       θ ≈53.1∘

Example 3: Word Problem

A ladder leans against a wall. The ladder is 5 m long and makes an angle of 

${60}^{\circ }$

$$\sin {60}^{\circ }=\frac{O}{H}$$

​

$$0.866=\frac{O}{5}$$$$O=5\times 0.866=4.33\text{m}$$

So the ladder reaches about 4.3 m high.

7. Practice Questions (20)

Section A: Basic Ratios

1. In a right triangle, 
   $\theta ={45}^{\circ }$
   , hypotenuse = 10 cm. Find opposite side.

2. If 
   $\sin \theta =\frac{3}{5}$
   ,  find 
   $\cos \theta$
   
   

3. A right triangle has adjacent side = 12 cm and hypotenuse = 13 cm. Find 
   $\cos \theta$
   
   

4. Find the value of 
   $\tan {30}^{\circ }$
   .

5. A right triangle has sides O = 8 cm, A = 15 cm. Find the hypotenuse using Pythagoras.

Section B: Missing Sides

6. Find the adjacent side if 
   $\theta ={60}^{\circ }$

7. A ladder of length 10 m makes an angle of 
   ${30}^{\circ }$
    with the ground. Find the height it reaches.

8. If 
   $\cos \theta =\frac{5}{13}$
    , find the adjacent side when H = 13.

9. A flagpole casts a shadow of 12 m when the angle of elevation of the top is 
   ${45}^{\circ }$
   . Find the height of the pole.

10. A ramp makes a 
    ${20}^{\circ }$
     angle with the ground. If the ramp is 5 m long, how high is it?

Section C: Missing Angles

11. In a right triangle, O = 7 cm, H = 25 cm. Find 
    $\theta$
    
    

12. Find 
    $\theta$
     if adjacent = 9 cm, hypotenuse = 15 cm.

13. A ladder 8 m long reaches 6 m up a wall. Find the angle of elevation.

14. A ramp 4 m long rises to 1 m height. Find the angle of elevation.

15. In a triangle, O = 24 cm, A = 7 cm. Find 
    $\theta$
    
    

Section D: Word Problems

16. A building is 50 m tall. A person standing 50 m away from the base looks at the top. Find the angle of elevation.

17. A boat at sea observes the top of a cliff at 
    ${30}^{\circ }$
    . If the cliff is 100 m tall, how far is the boat from the base?

18. A plane is flying at a height of 2,000 m. If the angle of depression to a point on the ground is 
    ${15}^{\circ }$
    , how far is the point horizontally?

19. A boy flying a kite holds the string at ground level. The string is 40 m long, and the angle of elevation is 
    ${60}^{\circ }$
    . How high is the kite?

20. A surveyor measures the angle of elevation to the top of a tower as 
    ${40}^{\circ }$
     from a distance of 30 m. Find the height of the tower.

We have finally come to the end of this lesson on Trigonometry. Hope to hear from you regarding the solutions.