Applications of Right-Angled Triangles Revision Notes for VCE SSCE General Mathematics

Applications of Right-Angled Triangles

Introduction

When you encounter real-world problems involving heights, distances, and angles, right-angled triangles provide a powerful tool for finding unknown measurements. This topic shows you how to use trigonometric ratios and their inverses to solve practical problems. The key skills you'll develop are drawing clear diagrams, identifying which sides and angles you know, and choosing the correct trigonometric function to solve for the unknown.

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The three essential skills for solving right-angled triangle problems are:

Drawing clear, labelled diagrams
Identifying which sides and angles are known
Selecting the appropriate trigonometric function

Finding unknown lengths

Use trigonometric ratios to find an unknown side length when you know one angle (other than the right angle) and one side length. This approach allows you to calculate distances and heights that would be difficult or impossible to measure directly.

Step-by-step process

Draw a clear diagram showing the right-angled triangle with all given information
Label the unknown side with a letter (usually $x$ )
Identify the sides relative to the given angle:
- Opposite: the side across from the angle
- Adjacent: the side next to the angle (not the hypotenuse)
- Hypotenuse: the longest side, opposite the right angle
Choose the appropriate trigonometric ratio based on which sides are involved:
- $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$
- $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$
- $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
Substitute the known values into the formula
Solve for the unknown by rearranging the equation
Calculate and round your answer appropriately

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Worked Example: Finding the height of a flagpole

Problem: A flagpole creates a shadow that measures 7.42 m along the ground. The sun's rays make an angle of 38° with the level ground. Calculate the height of the flagpole to two decimal places.

Solution:

Start by sketching a right-angled triangle representing this situation. Mark the known measurements and label the unknown height as $x$ .

Since we're working with the opposite side (height) and the adjacent side (shadow length), the tangent ratio is appropriate:

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

Substitute the known values:

$\tan 38° = \frac{x}{7.42}$

Multiply both sides by 7.42 to isolate $x$ :

$7.42 \times \tan 38° = x$

Use your calculator to evaluate this:

$x = 5.797...$

Rounding to two decimal places:

The height of the flagpole is 5.80 m.

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Practice problem: Walking up a slope

Problem: A person walks 200 m up a slope inclined at 35° to the horizontal. How much have they risen vertically? Answer to the nearest metre.

Hints:

The sloping distance (200 m) is the hypotenuse
The vertical rise is the opposite side to the 35° angle
Use the trigonometric ratio that connects opposite and hypotenuse

Finding unknown angles

Use inverse trigonometric functions to find an unknown angle when you know two side lengths of the right-angled triangle. This technique is particularly useful when you need to determine the angle of elevation, inclination, or slope in practical situations.

Step-by-step process

Draw a clear diagram showing the right-angled triangle with all given measurements
Label the unknown angle with a Greek letter (commonly $\theta$ )
Identify the known sides relative to the unknown angle:
- Opposite, adjacent, or hypotenuse
Choose the appropriate inverse function based on which sides you know:
- $\theta = \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$
- $\theta = \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right)$
- $\theta = \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$
Substitute the known values into the inverse function
Use your calculator to evaluate the inverse function
Round your answer to the required number of decimal places or nearest degree

lightbulbExample

Worked Example: Finding the angle of a sloping roof

Problem: A sloping roof uses sheets of corrugated iron 4.2 m long on a shed that is 4 m wide. The roof sheets don't overlap past the sides of the walls. Find the angle that the roof makes with the horizontal, to one decimal place.

Solution:

Begin by sketching the right-angled triangle that represents this situation. Mark the known lengths and label the required angle as $\theta$ .

Since we know the adjacent side (4 m) and the hypotenuse (4.2 m), the cosine ratio is the appropriate choice:

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

Substitute the known values:

$\cos \theta = \frac{4}{4.2}$

To find the angle, use the inverse cosine function:

$\theta = \cos^{-1}\left(\frac{4}{4.2}\right)$

Calculate this expression as a whole on your calculator:

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

$\cos \theta = \frac{4}{4.2}$

$\theta = \cos^{-1}\left(\frac{4}{4.2}\right)$

$\theta = 17.7528...°$

Rounding to one decimal place:

The roof makes an angle of 17.8° with the horizontal.

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Practice problem: Finding the angle of a slide

Problem: A vertical ladder of a children's slide is 2 m in height. The length of the slide itself is 3.5 m. Find the angle $\theta$ that the slide makes with the level ground, to the nearest degree.

Hints:

Identify which sides you know: the opposite (2 m) and hypotenuse (3.5 m)
Choose the inverse trigonometric function that uses these two sides

Important tips for solving problems

Draw diagrams first

Always begin by sketching a clear, labelled diagram. This helps you visualise the problem and identify which sides and angles are involved. A good diagram is the foundation of accurate problem-solving in trigonometry.

Label everything

Mark all known measurements on your diagram and clearly label the unknown quantity you're trying to find. This prevents confusion about which values to use in your calculations.

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Avoid rounding errors

Always evaluate a mathematical expression as a whole on your calculator, rather than breaking it into several smaller calculations. When you round intermediate answers and then use them in further calculations, rounding errors accumulate and your final answer becomes less accurate.

For example, calculate $\cos^{-1}(4/4.2)$ in one step, not as separate steps of "4 ÷ 4.2 = 0.95" then " $\cos^{-1}(0.95)$ ".

Check your answer makes sense

After calculating, ask yourself: "Is this answer reasonable?" For instance, if you're finding the height of a flagpole and get an answer of 500 m, something has likely gone wrong!

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Key Points to Remember:

For finding lengths: Choose the trigonometric ratio ( $\sin$ , $\cos$ , or $\tan$ ) that connects the known angle and sides with the unknown side.
For finding angles: Choose the inverse trigonometric function ( $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ ) that uses the two known sides.
Always start with a diagram: Draw and label a right-angled triangle showing all the given information before attempting calculations.
Use SOH-CAH-TOA: This helps you remember which ratio to use: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Minimize rounding errors: Calculate the entire expression in one step on your calculator rather than rounding intermediate values.

Applications of Right-Angled Triangles (VCE SSCE General Mathematics): Revision Notes