Trigonometric Ratios Revision Notes for HSC SSCE Mathematics Standard

Trigonometric Ratios

Introduction to trigonometric ratios

Trigonometric ratios are mathematical relationships that connect the angles and sides of right-angled triangles. We use three main trigonometric ratios: sine ( $\sin \theta$ ), cosine ( $\cos \theta$ ), and tangent ( $\tan \theta$ ). These ratios help us solve problems involving right-angled triangles and are fundamental tools in mathematics.

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Understanding trigonometric ratios is essential for solving a wide range of mathematical problems, from finding unknown sides and angles in triangles to applications in physics, engineering, and navigation.

Naming the sides of a right-angled triangle

Before working with trigonometric ratios, you need to identify the three sides of a right-angled triangle correctly. Each side has a specific name depending on its position relative to the angle you're working with.

The three sides are:

Hypotenuse: This is always the longest side of the triangle. It sits directly opposite the right angle (the 90° angle).
Opposite side: This side is opposite to the angle $\theta$ (theta) that you're considering. The angle theta is the Greek letter commonly used to represent an angle in mathematics.
Adjacent side: This is the side next to the angle $\theta$ , but it's not the hypotenuse. Think of it as the side that forms part of the angle along with the hypotenuse.

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Critical Concept: The opposite and adjacent sides can change depending on which angle you're using. If you consider the other acute angle in the triangle instead of $\theta$ , the labels for opposite and adjacent would swap positions. However, the hypotenuse always stays the same because it's always opposite the right angle.

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Worked Example: Identifying triangle sides

Let's say you have a right-angled triangle with sides measuring 3, 4, and 5, where $\theta$ is at one of the acute angles.

Hypotenuse = 5 (the longest side, opposite the right angle)
Opposite side = 4 (the side across from angle $\theta$ )
Adjacent side = 3 (the side next to angle $\theta$ , excluding the hypotenuse)

Remember: Always identify which angle you're working with first, then label the sides accordingly.

The three trigonometric ratios

Once you've correctly named the sides of your triangle, you can calculate the three trigonometric ratios. Each ratio compares two specific sides of the triangle.

Sine ratio

The sine of an angle is the ratio of the opposite side to the hypotenuse:

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

Cosine ratio

The cosine of an angle is the ratio of the adjacent side to the hypotenuse:

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

Tangent ratio

The tangent of an angle is the ratio of the opposite side to the adjacent side:

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

The SOH CAH TOA mnemonic

To help you remember which ratio uses which sides, use the mnemonic SOH CAH TOA (pronounced as one word, like "so-ka-toe-ah").

SOH: Sine - Opposite - Hypotenuse
CAH: Cosine - Adjacent - Hypotenuse
TOA: Tangent - Opposite - Adjacent

The order of the letters in each part matches the order in the ratio. For example, SOH tells you that sine equals opposite divided by hypotenuse.

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Exam tip: Write "SOH CAH TOA" at the top of your working when solving trigonometry problems. This simple step can help prevent mistakes and remind you of the correct ratios to use.

Understanding why trigonometric ratios are constant

A key property of trigonometric ratios is that they remain constant for a specific angle, regardless of the size of the triangle. Let's explore what this means.

Triangles with 30° angles

Consider three different-sized right-angled triangles, all containing a 30° angle. Although the triangles are different sizes, they all have the same sine ratio:

Small triangle: $\sin 30° = \frac{1}{2} = 0.5$
Medium triangle: $\sin 30° = \frac{2}{4} = 0.5$
Large triangle: $\sin 30° = \frac{3}{6} = 0.5$

This demonstrates that any right-angled triangle with a 30° angle will have a sine ratio of 0.5. If you change the angle, the ratio changes, but all triangles with the same angle share the same trigonometric ratios.

Triangles with 45° angles

Similarly, let's examine three right-angled triangles with 45° angles. For these triangles, the tangent ratio is always 1:

Small triangle: $\tan 45° = \frac{1}{1} = 1$
Medium triangle: $\tan 45° = \frac{2}{2} = 1$
Large triangle: $\tan 45° = \frac{3}{3} = 1$

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This consistency applies to all three trigonometric ratios (sine, cosine, and tangent). The size of the triangle doesn't matter; only the angle affects the ratio values.

Worked example: Finding all three trigonometric ratios

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Worked Example: Finding all three trigonometric ratios

Let's find the sine, cosine, and tangent ratios for angle $\theta$ in a triangle with sides 8, 15, and 17.

Step 1: Name the sides of the triangle

Adjacent side: $a = 15$
Opposite side: $o = 8$
Hypotenuse: $h = 17$

Step 2: Calculate the sine ratio using SOH

$\sin \theta = \frac{o}{h}$

$\sin \theta = \frac{8}{17}$

Step 3: Calculate the cosine ratio using CAH

$\cos \theta = \frac{a}{h}$

$\cos \theta = \frac{15}{17}$

Step 4: Calculate the tangent ratio using TOA

$\tan \theta = \frac{o}{a}$

$\tan \theta = \frac{8}{15}$

Therefore, for this triangle: $\sin \theta = \frac{8}{17}$ , $\cos \theta = \frac{15}{17}$ , and $\tan \theta = \frac{8}{15}$ .

Worked example: Finding a trigonometric ratio using Pythagoras' theorem

Sometimes you'll know one trigonometric ratio and need to find another. In these cases, you might need to use Pythagoras' theorem to find a missing side first.

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Worked Example: Finding a trigonometric ratio using Pythagoras' theorem

Question: Find $\sin \theta$ in simplest form given $\tan \theta = \frac{6}{8}$ .

Step 1: Draw a triangle and label the known sides

Since $\tan \theta = \frac{o}{a} = \frac{6}{8}$ , we know:

Opposite side = 6
Adjacent side = 8

Step 2: Use Pythagoras' theorem to find the hypotenuse

Pythagoras' theorem states: $h^2 = a^2 + o^2$

Step 3: Substitute the known values

$h^2 = 6^2 + 8^2$

Step 4: Calculate the hypotenuse

h = \sqrt{6^2 + 8^2}

h = \sqrt{36 + 64}

h = \sqrt{100}

h = 10

Step 5: Find the sine ratio

$\sin \theta = \frac{o}{h}$

$\sin \theta = \frac{6}{10}$

Step 6: Simplify the ratio

$\sin \theta = \frac{3}{5}$

Therefore, $\sin \theta = \frac{3}{5}$ .

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Exam tip: Always simplify your trigonometric ratios to their simplest form unless the question specifies otherwise. Divide both the numerator and denominator by their highest common factor.

Remember!

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Key Points to Remember:

Naming sides: The hypotenuse is always opposite the right angle. The opposite side is across from your angle $\theta$ , and the adjacent side is next to $\theta$ (but not the hypotenuse).
Three key ratios: $\sin \theta = \frac{o}{h}$ , $\cos \theta = \frac{a}{h}$ , and $\tan \theta = \frac{o}{a}$ . Use SOH CAH TOA to remember these.
Constant ratios: Trigonometric ratios stay the same for a given angle, no matter how large or small the triangle is.
Pythagoras' theorem: When you need to find a missing side before calculating a trigonometric ratio, use $h^2 = a^2 + o^2$ .
Always simplify: Express your final answers as simplified fractions where possible.

Trigonometric Ratios (HSC SSCE Mathematics Standard): Revision Notes