Trigonometry [Theory Note]

 Trigonometry

The word comes from the Greek words trigonon ("triangle") and metron ("measure").

Relationship between Angles and lengths of a triangle.

In a triangle , there are six data, 3 angles and 3 sides.

In Right triangle, if you know any 2 data out of 6, you can find any other missing data by trigonometry. 

1. Geometric Trigonometry (Right-Angled)

Do you know that?  

                            SOH CAH TOA

1. Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

2. Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

3. Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

What actually is O,A,H ?

O=opposite

A=adjacent

H=Hypotenuse

Hypotenuse is pretty clear, it is the side opposite of right angle.

Then what about adjacent or opposite?

It is from the point of view of the working angle, in the above drawing, Respect to $\theta$, the side touching $\theta$ is adjacent and the other is opposite.

Alright , let's move to general rules of all triangles (not just right triangle)

2. The Sine and Cosine Rules

When a triangle does not have a right angle, we use these two powerful rules. 

The Sine Rule

Use this when you have "pairs" of sides and angles.

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

The Cosine Rule

Use this for the "SAS" (Side-Angle-Side) or "SSS" (Side-Side-Side) scenarios.

• To find a side: $a^2 = b^2 + c^2 - 2bc \cos A$

• To find an angle: $\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

3. Area of Any Triangle

You no longer need to rely solely on $\frac{1}{2} \times \text{base} \times \text{height}$. If you know two sides and the included angle (SAS), use:

$$\text{Area} = \frac{1}{2}ab \sin C$$

Note that , the angle and side must be  SIDE-ANGLE--SIDE position.

Use 

a-$\gamma$-b

b-$\alpha$-c

c-$\beta$-a

4. Bearings and 3D Trigonometry

In IGCSE exams, trigonometry is often applied to real-world problems:

• Bearings: Always measured from North, in a clockwise direction, and written as three digits (e.g., 045°)

         If you are not sure about the starting point, find the word "from".

• 3D Trig: The trick here is to visualize 2D triangles "hidden" inside the 3D shape (like the diagonal of a cuboid or the slant height of a pyramid).

For 3D shapes, you can only practice and visualize as much as possible, there is no short cut. For me, I was very poor with those 3D shapes but now I am good enough to see the hidden ones. I normalize those shapes by studying many past papers.

Make sure your calculator is always in DEG (Degree) mode! Many marks are lost simply because a calculator was accidentally set to Radians or Gradians.

Not to forget to use Inverse, if the angle is like around decimals or the value is too low, check again.

Good luck.....