DoAssignment.ca

M7 · Describe how trigonometry is used in an occupation

Learn to describe how trigonometry is used in an occupation through clear examples and targeted practice.

Ontario Grade 10 Mathematics

Measurement and Trigonometry

How occupations use angles and side lengths to solve practical problems

Imagine a surveyor who needs to estimate the height of a building. Climbing to the roof with a measuring tape may be unsafe or impractical. The surveyor can measure a horizontal distance on the ground and the angle from that point to the top of the building. Trigonometry connects those measurements to the unknown height. In this lesson, you will see how occupations use that connection. The focus is on describing the use of trigonometry, not on memorizing job facts.

What you will learn

1. Bridge from right triangles

A right triangle has one angle that measures 90∘90^\circ. Its longest side, opposite the right angle, is called the hypotenuse. The other two sides are called legs. These words help us describe a triangle, but a real object or work site does not always have a triangle drawn on it.
Trigonometry is a way to connect an angle in a right triangle with the lengths of its sides. The angle is often measured with an instrument. A side length may be measured directly, or it may be the value a worker needs to find.
Before choosing a ratio, the worker must know which angle is being used. Relative to that angle, the opposite side is across from it. The adjacent side touches it and is not the hypotenuse. These names depend on the angle chosen.

2. The three side ratios

A ratio compares two quantities by division. For a right triangle, the sine, cosine, and tangent ratios each compare a particular pair of side lengths to an acute angle. An acute angle is greater than 0∘0^\circ and less than 90∘90^\circ.
For an angle called θ\theta (the Greek letter theta), sine compares the opposite side with the hypotenuse. Cosine compares the adjacent side with the hypotenuse. Tangent compares the opposite side with the adjacent side. The abbreviations are sin, cos, and tan.
A helpful memory phrase is SOH-CAH-TOA. It reminds you which sides go with each ratio. It is not a separate rule for choosing the angle: first label the triangle correctly, then select the ratio that connects the known and unknown sides.
In an occupation, these ratios are useful when direct measurement is difficult. A worker might measure an angle and one side, then use the matching ratio to determine another side. A calculator can evaluate the ratio for the measured angle. Workers must set the calculator to degrees when the angle measurement is given in degrees.
sin⁡θ=oppositehypotenuse,cos⁡θ=adjacenthypotenuse,tan⁡θ=oppositeadjacent\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}

3. Occupations and practical decisions

Different jobs use trigonometry for different purposes. A construction worker may need to check a roof slope or plan a ramp. A surveyor may estimate a distance or height that cannot be reached directly. A technician working with a tall structure may use measured angles and distances to help locate or describe a feature.
In each case, the worker first decides what needs to be known. Then the worker identifies measurements that can be taken safely and accurately. A right-triangle diagram helps show how those measurements relate. The matching ratio is chosen only after the sides and angle have been labelled.
Trigonometry is one part of a job, not a replacement for careful work. The triangle must represent the actual situation. The measurements need suitable tools, and the result may be an estimate because measurements have limited precision. Work that affects safety must follow the appropriate workplace procedures and be checked with suitable methods.

4. Guided example: estimating a building's height

A surveyor stands on level ground, 1818 m from a building. The angle from the horizontal ground line to the top of the building is 32∘32^\circ. Assume the ground is level and the building is vertical. The surveyor wants to estimate the building's height above the ground.
The horizontal ground distance and the vertical building height make the legs of a right triangle. The measured angle is at the surveyor. The building height is opposite that angle, and the ground distance is adjacent. Since the known and unknown sides are opposite and adjacent, tangent fits this situation.
tan⁡32∘=h18\tan 32^\circ=\frac{h}{18}

5. Independent practice and review

For each situation, describe what a worker could measure, what they want to find, and how a right-triangle model could help. You do not need to solve these prompts. A strong response names the occupation, the practical task, and the measurements or angle involved.
A roofing worker checks the slope of a roof. Which lengths or angle could be relevant? A surveyor needs to estimate the height of a tree without climbing it. What could be measured from a safe location? A worker planning a ramp needs to describe its steepness. How might the ramp and ground form part of a right triangle?
Use this short checklist when explaining an occupational use of trigonometry: identify the task; say what can be measured and what is unknown; describe the right triangle; name the angle and sides; and explain why a ratio could connect them. If a calculation is made, include the units and consider whether the assumptions make sense.

Examples of trigonometry in occupations

OccupationPossible taskHow trigonometry may help
SurveyorEstimate a height or distance that is hard to reachUse a measured angle and a known distance in a right-triangle model
Construction workerPlan or check a roof slope or rampRelate an angle to side lengths in the design
Technician working with a tall structureDescribe the location of a featureUse measured angles and distances as part of the model

Worked example

Estimating the height of a building

A surveyor stands on level ground, 1818 m from a vertical building. The angle from the surveyor's horizontal line of sight to the top is 32∘32^\circ. Estimate the building's height above the ground.
  1. Model the situation
    Represent the building height and the ground distance as the legs of a right triangle. The angle is at the surveyor. The height is opposite the angle, and the 1818 m ground distance is adjacent.
  2. Choose a ratio
    Tangent relates the opposite side to the adjacent side. Those are the two sides in this situation, so use tangent.
    tan⁡32∘=h18\tan 32^\circ=\frac{h}{18}
  3. Find the height
    Multiply both sides by 1818 to isolate hh. Evaluate the tangent with a calculator in degree mode, then round the estimate to the nearest tenth of a metre.
    h=18tan⁡32∘≈11.2 mh=18\tan 32^\circ\approx 11.2\text{ m}
Answer: The building is about 11.211.2 m high under the stated assumptions.
Check: The angle is less than 45∘45^\circ, so the opposite side should be shorter than the adjacent side. The estimate of 11.211.2 m is less than 1818 m, which is reasonable.

Common mistakes and how to avoid them

Choosing sine, cosine, or tangent before labelling the sides.
Correction: Identify the angle first. Label the opposite, adjacent, and hypotenuse sides from that angle, then choose the ratio that relates the known and unknown sides.
Calling any side the hypotenuse.
Correction: The hypotenuse is always opposite the right angle. It is not defined by where the worker is standing.
Treating a calculated value as exact even when measurements are estimates.
Correction: Describe the result as an estimate, include units, and state important assumptions such as level ground or a vertical structure.
Using a calculator set to radians when the angle is given in degrees.
Correction: Check the calculator's angle setting. Use degree mode for an angle measured in degrees.

Lesson summary

Check your understanding

Question 1

A worker knows the adjacent side and wants the opposite side relative to a measured angle. Which ratio directly connects those two sides?
  1. Sine
  2. Cosine
  3. Tangent
  4. None of the three
Show answer and explanation
Tangent
Tangent compares the opposite side with the adjacent side.

Question 2

A surveyor uses an angle and a measured ground distance to estimate a building height. What makes trigonometry useful here?
  1. It connects an angle and known side to another side in a right-triangle model.
  2. It removes the need to measure anything.
  3. It guarantees an exact result even if the measurements are rough.
  4. It changes the building into a right triangle.
Show answer and explanation
It connects an angle and known side to another side in a right-triangle model.
The triangle is a model of the situation. A ratio connects the measured angle and side to the unknown height.

Question 3

What should a worker do before choosing a trigonometric ratio?
  1. Label the angle and sides in the right-triangle model.
  2. Assume the longest side is adjacent.
  3. Choose a ratio at random and then label the sides.
  4. Ignore whether the angle is given in degrees.
Show answer and explanation
Label the angle and sides in the right-triangle model.
The side labels depend on the selected angle. Labelling first helps the worker choose the ratio that matches the known and unknown sides.

Key terms

Right triangle
A triangle with one angle measuring 90∘90^\circ.
Hypotenuse
The longest side of a right triangle, opposite the right angle.
Opposite side
The side across from the angle being used.
Adjacent side
The side beside the angle being used that is not the hypotenuse.
Ratio
A comparison of quantities by division.
Estimate
A value close to the actual value, often based on measured information.

Continue through MFM2P

View the complete MFM2P Ontario Grade 10 Mathematics curriculum and lessons

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic M7. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question