Finding an Unknown Side in a Right-Angled Triangle Revision Notes for VCE SSCE General Mathematics

Finding an Unknown Side in a Right-Angled Triangle

Introduction

When you know one angle and the length of one side in a right-angled triangle, you can calculate the length of another side using trigonometric ratios. This is a powerful technique that works for any right-angled triangle, regardless of its size or orientation.

The key is choosing the correct trigonometric ratio based on which sides are involved in your problem.

The three trigonometric ratios

There are three main trigonometric ratios you need to know: sine, cosine, and tangent. Each ratio relates two sides of a right-angled triangle to an angle.

Sine (sin)

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

Use sine when you're working with the opposite side and the hypotenuse.

Cosine (cos)

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

Use cosine when you're working with the adjacent side and the hypotenuse.

Tangent (tan)

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

Use tangent when you're working with the opposite side and the adjacent side.

Remembering the ratios: SOH-CAH-TOA

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The SOH-CAH-TOA Mnemonic

A helpful way to remember these ratios is the mnemonic SOH-CAH-TOA:

Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent

This mnemonic is one of the most useful memory aids in trigonometry!

Identifying the sides

Before using any trigonometric ratio, you need to identify the sides of your triangle:

Hypotenuse: Always the longest side, opposite the right angle
Opposite: The side across from the angle you're working with
Adjacent: The side next to the angle you're working with (but not the hypotenuse)

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Critical Point About Side Labels

Remember that the "opposite" and "adjacent" labels depend on which angle you're considering. They can change if you work with a different angle in the same triangle. Always identify these sides relative to the angle you're using, not the right angle.

Finding an unknown side in the numerator

When the unknown side appears in the top (numerator) of your trigonometric ratio, the solution method is straightforward.

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Worked Example: Finding the Opposite Side

Find the length of the unknown side, $x$ , in the triangle shown to two decimal places.

Step 1: Identify which sides are involved

Looking from the 38° angle:

The side we're finding ( $x$ ) is the opposite side
The side we know (65) is the hypotenuse

Step 2: Choose the appropriate trigonometric ratio

Since we're using the opposite and the hypotenuse, we use sine:

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

Step 3: Substitute the known values

$\sin 38° = \frac{x}{65}$

Step 4: Rearrange to solve for $x$

Multiply both sides by 65:

$65 \times \sin 38° = x$

$x = 65 \times \sin 38°$

Step 5: Calculate using a calculator

$x = 65 \times \sin 38°$

$x = 40.017...$

Step 6: Round to two decimal places

$x = 40.02$

Method summary for numerator cases

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Standard Method When Unknown is in the Numerator:

Identify the positions of the known and unknown sides relative to the given angle
Select the trigonometric ratio that uses both these sides
Substitute the known values into the ratio
Multiply both sides by the denominator to isolate $x$
Use your calculator to evaluate the expression
Round your answer to the required number of decimal places

Finding an unknown side in the denominator

When the unknown side appears in the bottom (denominator) of your trigonometric ratio, you need an extra step in your working.

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Worked Example: Finding the Hypotenuse

Find the value of $x$ in the triangle shown to two decimal places.

Step 1: Identify which sides are involved

Looking from the 34° angle:

The side we know (72) is the adjacent side
The side we're finding ( $x$ ) is the hypotenuse

Step 2: Choose the appropriate trigonometric ratio

Since we're using the adjacent and the hypotenuse, we use cosine:

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

Step 3: Substitute the known values

$\cos 34° = \frac{72}{x}$

Step 4: Multiply both sides by $x$

$x \cos 34° = 72$

Step 5: Divide both sides by $\cos 34°$

$x = \frac{72}{\cos 34°}$

Step 6: Calculate using a calculator

$x = \frac{72}{\cos 34°}$

$x = 86.847...$

Step 7: Round to two decimal places

$x = 86.85$

Method summary for denominator cases

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Modified Method When Unknown is in the Denominator:

Identify the positions of the known and unknown sides relative to the given angle
Select the trigonometric ratio that uses both these sides
Substitute the known values into the ratio
Multiply both sides by $x$ to remove it from the denominator
Divide both sides by the trigonometric function to isolate $x$
Use your calculator to evaluate the expression
Round your answer to the required number of decimal places

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Why the Extra Step?

When $x$ is in the denominator, you can't simply multiply both sides by the denominator because the denominator itself contains the unknown. Instead, you must first multiply by $x$ to move it to the numerator, then divide by the trigonometric value to isolate $x$ completely.

Complete method for finding unknown sides

Here's a systematic approach that works for any problem:

Sketch and label: Draw the triangle clearly and mark the given angle and known side length. Label the unknown side as $x$
Identify positions: Determine the position names (opposite, adjacent, hypotenuse) for both the known side and the unknown side, relative to the given angle
Choose the ratio: Select the trigonometric ratio that includes both the known and unknown sides:
- Use $\sin \theta$ for opposite and hypotenuse
- Use $\cos \theta$ for adjacent and hypotenuse
- Use $\tan \theta$ for opposite and adjacent
Write the equation: Substitute the known values into your chosen ratio
Rearrange: Solve the equation to make $x$ the subject:
- If $x$ is in the numerator, multiply both sides by the denominator
- If $x$ is in the denominator, first multiply both sides by $x$ , then divide both sides by the trigonometric value
Calculate: Use your calculator to find the numerical value
Round: Express your answer to the required number of decimal places

Calculator tips

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Using Your Calculator Effectively:

Make sure your calculator is in degree mode (not radians)
For expressions like $65 \times \sin 38°$ , enter: 65 × sin(38) =
For expressions like $\frac{72}{\cos 34°}$ , enter: 72 ÷ cos(34) =
Always write down more decimal places than required before rounding your final answer

Common mistakes to avoid

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Watch Out for These Common Errors:

Mixing up opposite and adjacent: Always identify these sides relative to the angle you're using, not the right angle
Forgetting to multiply by $x$ : When $x$ is in the denominator, remember you need to multiply both sides by $x$ before you can divide
Rounding too early: Keep extra decimal places during your calculation and only round at the final step
Wrong calculator mode: Ensure your calculator is set to degrees, not radians

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Key Points to Remember:

The three trigonometric ratios are: $\sin \theta = \frac{\text{opp}}{\text{hyp}}$ , $\cos \theta = \frac{\text{adj}}{\text{hyp}}$ , $\tan \theta = \frac{\text{opp}}{\text{adj}}$ (remember SOH-CAH-TOA!)
Always identify the position names (opposite, adjacent, hypotenuse) of the sides relative to the given angle before choosing your trigonometric ratio
When the unknown side is in the numerator, multiply both sides by the denominator to isolate it
When the unknown side is in the denominator, multiply both sides by the unknown first, then divide by the trigonometric value
Always check your calculator is in degree mode and round only your final answer to the required number of decimal places

Finding an Unknown Side in a Right-Angled Triangle (VCE SSCE General Mathematics): Revision Notes