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C1.5 · Solve real-world acute-triangle problems

Learn to solve real-world acute-triangle problems through clear examples and targeted practice.

Ontario Grade 11 Mathematics

Trigonometric Functions

Represent the situation, choose a law, and check the result

A triangle can model distances between places or points that are hard to measure directly. An acute triangle has three angles, each less than 90∘90^\circ. To solve a real-world problem, turn the story into a labelled triangle. Then choose a rule that connects what is known to what is unknown. This lesson reviews useful triangle facts, explains the sine law and cosine law, and uses them to find a distance.

What you will learn

1. Prerequisite bridge: read and label the triangle

A triangle has three sides and three interior angles. An acute angle is greater than 0∘0^\circ and less than 90∘90^\circ. The interior angles of every triangle add to 180∘180^\circ. These facts help you label a diagram and check a result.
Use capital letters for the angles and the matching lowercase letters for the opposite sides. For example, side aa is across from angle AA. The word opposite means across from, not next to. Matching each side with its opposite angle is important when using the sine law.
A sketch does not need to be drawn to scale. Its job is to show how the measurements fit together. Label the measurements from the story, and mark the unknown with a variable. If the measurements use different units, convert them to one unit before calculating. For example, change kilometres to metres if the other distances are in metres.
When an angle is given in degrees, make sure your calculator is set to degree mode. Follow the calculator instructions for entering the angle and the trigonometric function.
A+B+C=180∘A+B+C=180^\circ

2. Choose the law that fits the information

The sine law relates the length of a side to the sine of its opposite angle. The sine of an angle is a value found using a calculator. Look for a known side and its opposite angle. That matching pair can help you find another side or angle.
The cosine law relates three sides to one angle. It is useful when you know two sides and the angle between them. That angle is called the included angle. The cosine law can also find an angle when all three side lengths are known.
Before choosing a law, list the known measurements and identify what is unknown. Then check how the measurements are arranged in your triangle. Having an angle in the problem does not automatically mean the sine law is the right choice. If the angle is between two known sides, the cosine law is a direct fit.
Use enough calculator digits during the calculation. Round only the final result, unless the question asks for a different level of accuracy. A final measurement should have units and should answer the question in the story.
asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}

3. Represent the situation and check the answer

A clear solution has three connected parts: a labelled sketch, an equation, and a sentence that interprets the answer. The sketch translates the story into a triangle. The equation shows how the known measurements are related. The sentence tells what the calculated value means and gives its units.
For example, two distances from a landmark to two survey points form two sides of a triangle. If the angle between those distances is known, the unknown distance between the survey points is opposite that angle. This is a two-sides-and-included-angle situation, so use the cosine law.
Once you calculate, check whether the result fits the situation. A side length must be positive. The angles in an acute triangle must each be less than 90∘90^\circ and must add to 180∘180^\circ. Compare the size of the answer with the given distances. A result that is much too large or too small may signal a setup or calculator error.
Real measurements are often approximate. Your answer should not suggest more precision than the measurements support. If the question asks for the nearest metre, keep calculator values during the work and round the final distance to the nearest metre.
a2=b2+c2−2bccos⁡Aa^2=b^2+c^2-2bc\cos A

4. Apply the process to a surveying situation

A surveyor may be unable to measure the distance between two points directly. Instead, the surveyor can measure from each point to a landmark and measure the angle where the two sight lines meet. Those three pieces of information form a triangle. The cosine law finds the distance between the two points.
For a different arrangement, a known side and its opposite angle may be available. In that case, the sine law can connect that pair to another side-angle pair. The diagram helps you decide which arrangement you have; the law is not chosen just because it appears familiar.
A final check is part of solving the problem, not an extra calculation. Confirm that the equation matches the labelled triangle, that the calculator used degrees, and that the answer makes sense in the real setting. Then report the requested quantity in a complete sentence.
c2=a2+b2−2abcos⁡Cc^2=a^2+b^2-2ab\cos C

Worked example

Find the distance between two survey points

A surveyor stands at two points, PP and QQ. A landmark is at RR. The distances PRPR and QRQR are 120120 m and 9595 m. The angle between the two sight lines at RR is 68∘68^\circ. Find the distance PQPQ to the nearest metre.
  1. Sketch and identify
    Draw triangle PQRPQR. The two known sides meet at RR, and the unknown distance PQPQ is opposite the known angle there. This is a two-sides-and-included-angle situation.
    PR=120 m,QR=95 m,∠R=68∘PR=120\text{ m},\quad QR=95\text{ m},\quad \angle R=68^\circ
  2. Choose the cosine law
    The cosine law connects the two known sides and the angle between them to the opposite side. Let PQ=xPQ=x.
    x2=1202+952−2(120)(95)cos⁡68∘x^2=120^2+95^2-2(120)(95)\cos 68^\circ
  3. Calculate
    Evaluate the expression in degree mode. Take the positive square root because a distance cannot be negative. The result is approximately 122.0122.0 m before rounding.
    x≈122.0 mx\approx122.0\text{ m}
  4. Interpret and check
    The distance is positive and is less than the sum of the two distances through the landmark. To the nearest metre, the distance between the survey points is 122122 m.
    PQ≈122 mPQ\approx122\text{ m}
Answer: The distance between PP and QQ is approximately 122122 m.
Check: With the same two side lengths, an angle of 68∘68^\circ gives a shorter opposite side than a right angle would. The calculated distance is also less than 120+95=215120+95=215 m, so it is reasonable.

Common mistakes and how to avoid them

Using the sine law just because the problem gives an angle.
Correction: Check whether the angle is between two known sides. If it is, use the cosine law. Use the sine law when a known side and its opposite angle form a pair.
Matching a side with an angle that is beside it rather than opposite it.
Correction: Label the triangle so each lowercase side letter matches the capital letter of the angle across from it.
Using a calculator setting that does not match the angle units.
Correction: When the problem gives degrees, set the calculator to degree mode before evaluating a trigonometric expression.
Rounding during the calculation or leaving off units.
Correction: Keep calculator values until the final step. Round as requested and include the correct units in the answer.

Lesson summary

Check your understanding

Question 1

Two sides of an acute triangle are 77 cm and 1010 cm. The angle between them is 42∘42^\circ. Which method directly finds the third side?
  1. The sine law
  2. The cosine law
  3. Add the two side lengths
  4. Subtract the shorter side length from the longer side length
Show answer and explanation
The cosine law
The known angle is between the two known sides. The cosine law relates those measurements to the third side.

Question 2

In a labelled triangle, side cc is opposite which angle?
  1. Angle AA
  2. Angle BB
  3. Angle CC
  4. The largest angle only
Show answer and explanation
Angle CC
The lowercase side letter matches the capital letter of its opposite angle, so side cc is opposite angle CC.

Question 3

A triangle has angles 48∘48^\circ and 67∘67^\circ. What is its third angle?
  1. 55∘55^\circ
  2. 65∘65^\circ
  3. 75∘75^\circ
  4. 115∘115^\circ
Show answer and explanation
65∘65^\circ
Triangle angles add to 180∘180^\circ. The third angle is 180∘−48∘−67∘=65∘180^\circ-48^\circ-67^\circ=65^\circ.

Key terms

Acute angle
An angle greater than 0∘0^\circ and less than 90∘90^\circ.
Opposite side
The side across from a given angle.
Included angle
The angle between two specified sides.
Sine law
A rule that relates each side of a triangle to the sine of its opposite angle.
Cosine law
A rule that relates the sides of a triangle to the cosine of one of its angles.

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About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation C1.5. It is a study resource, not an official curriculum publication.

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