Solving Practical Problems Revision Notes for HSC SSCE Mathematics Standard

Solving Practical Problems

Introduction to practical trigonometry

Trigonometry helps us solve real-world measurement problems that would be difficult or impossible to measure directly. Common applications include:

Calculating the height of tall objects like trees or buildings
Determining the height of mountains
Finding the width of rivers
Measuring angles in construction projects

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When solving trigonometric problems in practical situations, always read questions thoroughly and create a clear diagram showing all the given information.

Step-by-step problem-solving approach

Follow these five essential steps when solving any trigonometric word problem:

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The Five-Step Approach:

Step 1: Read carefully

Read the question and underline key information
Identify what you need to find

Step 2: Draw a diagram

Sketch a right-angled triangle
Label all given measurements
Mark the unknown value with a symbol (like $x$ or $\theta$ )

Step 3: Use trigonometry

Choose the appropriate trigonometric ratio
Set up and solve the equation

Step 4: Check your work

Verify the answer is reasonable
Ensure units are correct

Step 5: Write a complete answer

Express your answer in words
Confirm you have answered the question asked

Finding the length of a side

When you need to find an unknown side length in a right-angled triangle, use the appropriate trigonometric ratio based on what information you have.

Worked example: Tent pole height

A vertical tent pole is supported by a rope tied to the top of the pole and to a peg on the ground. The rope is $3$ m in length and makes an angle of $29°$ to the horizontal. What is the height of the tent pole? Answer correct to two decimal places.

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Worked Example: Finding the Height of a Tent Pole

1. Draw and label the diagram

Let the height of the tent pole be $x$ metres.

2. Name the sides

From the angle of $29°$ :

Adjacent side: base (along the ground)
Opposite side: $x$ (the height we need)
Hypotenuse: $3$ m (the rope length)

3. Determine the ratio

Since we know the hypotenuse and need the opposite side, use the sine ratio (SOH):

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

4. Substitute known values

$\sin 29° = \frac{x}{3}$

5. Solve for $x$

Multiply both sides by $3$ :

$x = 3 \times \sin 29°$

6. Calculate

Using a calculator: $3 \times \sin 29° =$

$x = 1.454428...$

7. Round to two decimal places

$x \approx 1.45$

8. Write the answer in words

Therefore, the height of the tent pole is $1.45$ m.

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Exam tip: Always check your answer makes sense. Here, the height ( $1.45$ m) is less than the rope length ( $3$ m), which is correct for a $29°$ angle.

Finding an angle

When you need to find an unknown angle in a right-angled triangle, use inverse trigonometric functions.

Trigonometry has many practical applications in building and construction. Vertical parts of structures make right angles with horizontal parts. Sloping lines complete right-angled triangles, allowing us to calculate angles and side lengths.

Worked example: Roof angle

The sloping roof of a shed uses sheets of Colorbond steel $4.5$ m long on a shed $4$ m wide. There is no overlap of the roof past the sides of the walls. Find the angle the roof makes with the horizontal. Answer correct to the nearest degree.

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Worked Example: Finding the Roof Angle

1. Draw and label the diagram

Let the required angle be $\theta$ .

2. Name the sides

From the angle $\theta$ :

Adjacent side: $4$ m (width of shed)
Opposite side: height (unknown)
Hypotenuse: $4.5$ m (length of roof sheet)

3. Determine the ratio

Since we know the adjacent side and hypotenuse, use the cosine ratio (CAH):

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

4. Substitute known values

$\cos \theta = \frac{4}{4.5}$

5. Make $\theta$ the subject

Use the inverse cosine function:

$\theta = \cos^{-1}\left(\frac{4}{4.5}\right)$

6. Calculate

Using a calculator: SHIFT $\cos^{-1}$ $(4 \div 4.5)$ = or exe

$\theta = 27.26604445...$

7. Round to the nearest degree

$\theta \approx 27°$

8. Write the answer in words

Therefore, the roof makes an angle of $27°$ with the horizontal.

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Exam tip: To find an angle, you must use an inverse trigonometric function ( $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ ). Make sure you know how to access these on your calculator.

Choosing the correct ratio

To select the appropriate trigonometric ratio:

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Selecting the Right Ratio:

Identify which sides you know or need to find
Match these to the ratio:
- SOH: Use $\sin \theta = \frac{o}{h}$ when you have opposite and hypotenuse
- CAH: Use $\cos \theta = \frac{a}{h}$ when you have adjacent and hypotenuse
- TOA: Use $\tan \theta = \frac{o}{a}$ when you have opposite and adjacent
To find a side: substitute values and solve
To find an angle: use the inverse function ( $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ )

Remember!

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Key Points to Remember:

Always draw a clear, labelled diagram for practical trigonometry problems
Follow the five-step problem-solving approach: read, draw, calculate, check, answer
Use SOH CAH TOA to choose the correct trigonometric ratio based on which sides you know
To find a side length, use the trigonometric ratio directly
To find an angle, use inverse trigonometric functions ( $\sin^{-1}$ , $\cos^{-1}$ , or $\tan^{-1}$ )
Check your answer is reasonable and includes the correct units
Always write your final answer in words to ensure you've answered the question

Solving Practical Problems (HSC SSCE Mathematics Standard): Revision Notes