To solve triangle problems in A-Level Mathematics, we often use inverse trigonometric functions.

These are special functions like $\sin^{-1}(x)$, $\cos^{-1}(x)$, and $\tan^{-1}(x)$.

They help us find angles when we know the lengths of the sides.

This is really handy for both right-angled triangles and non-right-angled triangles by using the laws of sine and cosine.

Example: Right-Angled Triangle

Let’s say you have a right-angled triangle.

You know one angle, which is $\theta = 30^\circ$, and the length of the opposite side is 3 units.

To find the length of the hypotenuse, we can use the sine function:

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \implies \sin(30^\circ) = \frac{3}{h}$$

If we rearrange that, we get:

$$h = \frac{3}{\sin(30^\circ)} = \frac{3}{0.5} = 6 \text{ units}$$

Now, what if we only knew the side lengths instead and needed to find the angle?

In that case, we would use the inverse sine function:

$$\theta = \sin^{-1}\left(\frac{3}{h}\right) = \sin^{-1}\left(\frac{3}{6}\right) = \sin^{-1}(0.5) = 30^\circ$$

Example: Non-Right-Angled Triangle

Now, let’s look at a non-right-angled triangle.

Imagine you know two sides and the angle between them.

We can call these sides $a$ and $b$, and the angle between them $\theta$.

To find the third side $c$, we can use the cosine rule:

$$c^2 = a^2 + b^2 - 2ab \cos(\theta)$$

If you want to find the angle $C$ that is opposite side $c$, you can rearrange this to use the inverse cosine function:

$$C = \cos^{-1}\left(\frac{a^2 + b^2 - c^2}{2ab}\right)$$

Conclusion

Inverse trigonometric functions are really helpful when solving triangle problems.

They allow us to figure out unknown angles easily.

It’s important to remember when to use these functions—whether you have side lengths and need angles, or if you have angles and need to find sides.

This skill will be very useful as you continue learning in A-Level mathematics!