Finding an Unknown Side or Angle Revision Notes for HSC SSCE Mathematics Standard

Finding an Unknown Side or Angle

Introduction

Trigonometric ratios allow us to calculate unknown sides and angles in right-angled triangles. When you know at least one angle and one side, you can find an unknown side. When you know at least two sides, you can find an unknown angle. The key to solving these problems is using the SOH CAH TOA mnemonic and carefully labelling the triangle's sides.

infoNote

When to use which method:

If you know one angle and one side → find an unknown side
If you know two sides → find an unknown angle

Finding an unknown side

When you need to calculate the length of an unknown side in a right-angled triangle, trigonometric ratios are your tool of choice. This method works when you already know one angle (besides the right angle) and at least one side length.

Step-by-step procedure

Follow these steps systematically to find an unknown side:

Step 1: Label the sides of your triangle relative to the angle you're working with:

$h$ for hypotenuse (the longest side, opposite the right angle)
$o$ for opposite (the side across from your angle)
$a$ for adjacent (the side next to your angle, excluding the hypotenuse)

Step 2: Identify which trigonometric ratio to use by examining which sides you know and which you need to find. Use SOH CAH TOA to help:

SOH: $\sin \theta = \frac{o}{h}$ (Sine = Opposite / Hypotenuse)
CAH: $\cos \theta = \frac{a}{h}$ (Cosine = Adjacent / Hypotenuse)
TOA: $\tan \theta = \frac{o}{a}$ (Tangent = Opposite / Adjacent)

Step 3: Rearrange the equation algebraically to make the unknown side $x$ the subject.

Step 4: Use your calculator to evaluate the expression. Always check your calculator is in degree mode, not radian mode.

Step 5: Round your answer to the required level of accuracy (usually decimal places).

chatImportant

Critical: The labels "opposite" and "adjacent" change depending on which angle you're considering. Always label your triangle relative to the specific angle you're working with.

Worked example: Using tangent

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Worked Example: Using Tangent

Question: Find the length of the unknown side $x$ in the triangle shown. Answer correct to three decimal places.

Solution:

Name the sides: $a = 20$ , $o = x$ , $h$ (not given)
Determine the ratio: We have the adjacent and need the opposite, so use TOA

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

Substitute the known values:

$\tan 35° = \frac{x}{20}$

Multiply both sides by $20$ :

$x = 20 \times \tan 35°$

Calculate using your calculator: Press $20$ $\tan$ $35$ $=$

$x = 14.004150...$

Round to three decimal places:

$x \approx 14.004$

Worked example: Using sine

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Worked Example: Using Sine

Question: Find the length of the unknown side $x$ in a triangle where the hypotenuse is $25$ and the angle is $34°$ . Answer correct to two decimal places.

Solution:

Name the sides: $a$ (not given), $o = x$ , $h = 25$
Determine the ratio: We have the hypotenuse and need the opposite, so use SOH

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

Substitute the known values:

$\sin 34° = \frac{x}{25}$

Multiply both sides by $25$ :

$x = 25 \times \sin 34°$

Calculate: Press $25$ $\sin$ $34$ $=$

$x = 13.979822...$

Round to two decimal places:

$x \approx 13.98$

Finding an unknown side in the denominator

Sometimes the unknown side $x$ appears in the denominator of your trigonometric ratio. This happens when you're looking for the hypotenuse or the adjacent side in certain configurations. This requires an additional algebraic step to solve.

The extra algebraic step

When your unknown appears in the denominator, you'll get an equation like:

$\sin 60° = \frac{2}{x}$

chatImportant

Critical Algebraic Steps:

To solve when the unknown is in the denominator:

First, multiply both sides by $x$ to eliminate it from the denominator:

$x \times \sin 60° = 2$

Then, divide both sides by the trigonometric expression to isolate $x$ :

$x = \frac{2}{\sin 60°}$

Common mistake: Don't try to take the reciprocal directly - it's more error-prone. Always follow these two steps in order.

Worked example: Finding the hypotenuse

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Worked Example: Finding the Hypotenuse

Question: Find the length of the unknown side $x$ in the triangle shown. Answer correct to two decimal places.

Solution:

Name the sides: $a = 12$ , $o$ (not given), $h = x$
Determine the ratio: We have the adjacent and need the hypotenuse, so use CAH

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

Substitute the known values:

$\cos 40° = \frac{12}{x}$

Multiply both sides by $x$ :

$x \times \cos 40° = 12$

Divide both sides by $\cos 40°$ :

$x = \frac{12}{\cos 40°}$

Calculate: Press $12$ $\div$ $\cos$ $40$ $=$

$x = 15.66488747$

Round to two decimal places:

$x \approx 15.66$

infoNote

Exam tip: When the unknown is in the denominator, always multiply by the unknown first, then divide by the trig expression. This systematic approach reduces calculation errors.

Finding an unknown angle

When you know two sides of a right-angled triangle but need to find an angle, you use inverse trigonometric functions. These are written as $\sin^{-1}$ , $\cos^{-1}$ , and $\tan^{-1}$ (also called arcsin, arccos, and arctan).

infoNote

Understanding Inverse Functions:

The inverse trigonometric functions "undo" the regular trig functions. If $\sin \theta = 0.5$ , then $\theta = \sin^{-1}(0.5)$ . On your calculator, you typically access these by pressing SHIFT or 2nd before the trig button.

Step-by-step procedure

Step 1: Label the sides of your triangle: $h$ for hypotenuse, $o$ for opposite, $a$ for adjacent.

Step 2: Identify which trigonometric ratio to use based on the two sides you know. The SOH CAH TOA mnemonic still applies.

Step 3: Rearrange the equation to make the angle $\theta$ the subject using the inverse trigonometric function.

Step 4: Use your calculator to find $\theta$ . Ensure your calculator is in degree mode.

Step 5: Express your answer to the required accuracy (degrees, minutes, or decimal places).

Worked example: Finding an angle to the nearest degree

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Worked Example: Finding an Angle to the Nearest Degree

Question: Find the angle $\theta$ in a triangle where the hypotenuse is $24$ and the opposite side is $16$ . Give your answer to the nearest degree.

Solution:

Name the sides: $h = 24$ , $o = 16$ , $a$ (not given)
Determine the ratio: We have the opposite and hypotenuse, so use SOH

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

Substitute the known values:

$\sin \theta = \frac{16}{24}$

Make $\theta$ the subject using the inverse sine function:

$\theta = \sin^{-1}\left(\frac{16}{24}\right)$

Calculate: Press SHIFT $\sin^{-1}$ $(16 \div 24)$ $=$

or Press SHIFT $\sin^{-1}$ $16$ $\frac{a}{b}$ $24$ $=$

$\theta = 41.8103149...°$

Round to the nearest degree:

$\theta = 42°$

Worked example: Finding an angle to the nearest minute

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Worked Example: Finding an Angle to the Nearest Minute

Question: Find the angle $\theta$ in a triangle where the hypotenuse is $6.8$ and the adjacent side is $4.2$ . Give your answer to the nearest minute.

Solution:

Name the sides: $h = 6.8$ , $o$ (not given), $a = 4.2$
Determine the ratio: We have the adjacent and hypotenuse, so use CAH

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

Substitute the known values:

$\cos \theta = \frac{4.2}{6.8}$

Make $\theta$ the subject:

$\theta = \cos^{-1}\left(\frac{4.2}{6.8}\right)$

Calculate: Press SHIFT $\cos^{-1}$ $(4.2 \div 6.8)$ $=$

$\theta = 51.855486...°$

Convert to degrees and minutes and round to the nearest minute:

$\theta \approx 51°51'$

infoNote

Understanding minutes: There are 60 minutes in one degree. To convert the decimal part to minutes, multiply by $60$ . For example, $0.855486 \times 60 \approx 51$ minutes.

Summary

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Key Points to Remember:

SOH CAH TOA is your essential tool: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
When finding an unknown side: Use the standard trig ratio, rearrange to make $x$ the subject, then calculate.
When the unknown is in the denominator: Multiply both sides by $x$ first, then divide by the trig expression.
When finding an unknown angle: Use inverse trig functions ( $\sin^{-1}$ , $\cos^{-1}$ , $\tan^{-1}$ ) to make $\theta$ the subject.
Always check your calculator is in degree mode before calculating.
Label your triangle sides carefully relative to the angle you're working with - the opposite and adjacent change depending on which angle you're considering.

Finding an Unknown Side or Angle (HSC SSCE Mathematics Standard): Revision Notes

Finding an Unknown Side or Angle

Introduction

Finding an unknown side

Step-by-step procedure

Worked example: Using tangent

Worked example: Using sine

Finding an unknown side in the denominator

The extra algebraic step

Worked example: Finding the hypotenuse

Finding an unknown angle

Step-by-step procedure

Worked example: Finding an angle to the nearest degree

Worked example: Finding an angle to the nearest minute

Summary

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