Trigonometry Basics (VCE SSCE General Mathematics): Revision Notes

Trigonometry basics

Understanding trigonometry is essential for solving problems involving right-angled triangles. This note reviews the fundamental concepts you need to work confidently with trigonometric ratios. You may have studied some of this material before, but it's important to ensure you have a solid foundation.

Naming the sides of a right-angled triangle

In any right-angled triangle, we can identify three distinct sides. The names we give these sides depend on their position relative to the angle we're working with, which we typically label as $\theta$ (the Greek letter theta).

Understanding how to correctly identify each side is crucial for applying trigonometric ratios:

The hypotenuse represents the longest side in a right-angled triangle. It's always positioned opposite the right angle (the $90°$ angle). No matter which angle $\theta$ you're considering, the hypotenuse never changes - it's always the longest side opposite the right angle.

The opposite side sits directly across from angle $\theta$. This side changes depending on which angle you're examining. If you were to move $\theta$ to a different corner of the triangle, a different side would become the opposite side.

The adjacent side lies next to angle $\theta$ and forms one of the arms of this angle. It runs from $\theta$ to the right angle, but it's not the hypotenuse. Like the opposite side, which side is "adjacent" depends on the position of $\theta$.

The trigonometric ratios

Trigonometric ratios provide a way to relate the angles in a right-angled triangle to the lengths of its sides. There are three fundamental trigonometric ratios you need to know: sine, cosine, and tangent. We write these as $\sin \theta$, $\cos \theta$, and $\tan \theta$.

Each ratio represents a specific relationship between two sides of the triangle:

Sine relates the opposite side to the hypotenuse:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{o}{h}$

Cosine relates the adjacent side to the hypotenuse:

$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{h}$

Tangent relates the opposite side to the adjacent side:

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{o}{a}$

The SOH-CAH-TOA mnemonic

Students often use the mnemonic SOH-CAH-TOA to remember these three trigonometric ratios. Let's break down what this means:

• SOH reminds us that Sine equals Opposite over Hypotenuse
• CAH reminds us that Cosine equals Adjacent over Hypotenuse
• TOA reminds us that Tangent equals Opposite over Adjacent

Understanding what trigonometric ratios mean

When you use a calculator to find, for example, that $\sin 30° = 0.5$, what does this actually tell us? This value means that in all right-angled triangles containing a $30°$ angle, the ratio of the opposite side to the hypotenuse is always $0.5$, regardless of the triangle's size.

The diagram above shows three different right-angled triangles, each with a $30°$ angle. Notice that:

• In the first triangle: $\frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2} = 0.5$
• In the second triangle: $\frac{\text{opposite}}{\text{hypotenuse}} = \frac{2}{4} = 0.5$
• In the third triangle: $\frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{6} = 0.5$

Even though these triangles are different sizes, the ratio remains constant at $0.5$. This is the fundamental principle of trigonometry - the ratios depend only on the angle, not on the size of the triangle.

Similarly, for any right-angled triangle with a $30°$ angle:

$\cos 30° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{3}}{2} = 0.8660 \text{ (to four decimal places)}$

$\tan 30° = \frac{\text{opposite}}{\text{adjacent}} = \frac{1}{\sqrt{3}} = 0.5774 \text{ (to four decimal places)}$

These values are always the same for a $30°$ angle, no matter what size triangle you're working with.

Using a calculator to find trigonometric values

Your CAS calculator can quickly find the value of any trigonometric ratio for any angle. However, there's one critical setting you must check first.