M6 Solve real measurement problems with right-triangle trigonometry

Ontario Grade 10 Mathematics

Measurement and Trigonometry

Choose a ratio, match the sides, and interpret the result

A surveyor may need to estimate the height of a tree without climbing it. A right triangle can connect the distance from the tree to an angle of sight and the tree’s height. Right-triangle trigonometry gives a way to calculate a missing side when an angle and another side are known. This lesson reviews how to recognize the sides, choose a trigonometric ratio, and use the answer in a real situation.

What you will learn

Identify the sides of a right triangle in relation to a chosen angle.
Choose sine, cosine, or tangent to find a missing side.
Use right-triangle trigonometry to solve a real measurement problem.
Check that a calculated measurement makes sense and report it with units.

1. Grade 9 bridge: recognize the triangle and its sides

A triangle is a shape with three sides and three angles. A right triangle has one angle of exactly

90^\circ

. The side opposite that right angle is called the hypotenuse. It is always the longest side.

The other two sides are named in relation to the angle you are using. Choose one of the triangle’s acute angles, meaning an angle smaller than

90^\circ

. The side across from that angle is the opposite side. The side that touches the angle and is not the hypotenuse is the adjacent side.

The names opposite and adjacent can change if you choose the triangle’s other acute angle. The hypotenuse does not change. Before calculating, mark the angle you are using and label the sides from that angle’s point of view.

In a measurement problem, draw a simple sketch. Show the right angle, the known angle, the known length, and the length you need. The sketch does not need to be perfectly to scale. It helps connect the words in the problem to the sides of the triangle.

Find the right angle first; the side across from it is the hypotenuse.
Label opposite and adjacent using the specific angle in the problem.
A clear sketch can reveal which measurement is missing.

2. Plain language: choose a trigonometric ratio

A trigonometric ratio compares the lengths of two sides in a right triangle. The three ratios used here are sine, cosine, and tangent. Their names are often shortened to sin, cos, and tan. The letters in SOH CAH TOA help you remember which sides each ratio compares.

Sine uses opposite and hypotenuse. Cosine uses adjacent and hypotenuse. Tangent uses opposite and adjacent. Choose the ratio that includes both the side you know and the side you need. This avoids using a side that is not part of the question.

An angle must be entered in degrees for the examples in this lesson. When using a calculator, check that it is set to degree mode. The calculator’s sine, cosine, and tangent buttons find the ratio for an angle.

To find a missing side, write the appropriate ratio, put the known angle and side into it, and rearrange the equation to isolate the unknown side. Rearranging means using the same operation on both sides of an equation to leave the wanted quantity by itself.

A real answer needs a sensible unit and level of precision. If the measurements are in metres, report the result in metres. Rounding to the nearest tenth is often suitable when the measurements are given to that level. Keep extra digits during the calculation and round only the final answer.

\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}},\quad \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}},\quad \tan\theta=\frac{\text{opposite}}{\text{adjacent}}

Sine compares opposite with hypotenuse.
Cosine compares adjacent with hypotenuse.
Tangent compares opposite with adjacent.
Use the ratio that contains the known side and the unknown side.

3. Visual guide: connect the situation to the ratio

The table gives a quick way to choose a ratio. First decide whether the missing side is opposite, adjacent, or the hypotenuse. Then find the row that also includes the known side. The remaining side in that row is not needed.

For example, if you know an angle and the adjacent side and need the opposite side, the tangent ratio uses exactly those two sides. If you know the angle and the hypotenuse and need the adjacent side, cosine is the matching ratio.

In a real scene, the triangle may be imagined rather than drawn on the ground. A horizontal distance and a vertical height meet at a right angle. A line of sight can form the sloping side. The angle of elevation is the angle measured upward from a horizontal line to the line of sight. It can be used as the marked acute angle in the triangle.

Match the sides before selecting the ratio.
An angle of elevation is measured upward from horizontal.
A horizontal distance and vertical height form the perpendicular sides in many height problems.

4. Guided example and independent practice

Read the example and follow how the real measurements become a right triangle. The line of sight is the hypotenuse. The tree’s height is opposite the angle of elevation, and the ground distance is adjacent. Since the known length is adjacent and the unknown height is opposite, tangent is the useful ratio.

After the example, try the practice questions on your own. For each one, sketch and label a right triangle, choose a ratio, calculate, and include units. Estimate first when possible: a missing side should be a positive length, and the hypotenuse should be longer than either other side.

Translate the situation into side labels before calculating.
Rearrange the ratio to make the unknown side the subject.
Check the answer against the sketch and the measurements.

Choosing a ratio from the sides

Ratio	Sides compared	Useful when
Sine	Opposite and hypotenuse	The known and missing sides are opposite and hypotenuse
Cosine	Adjacent and hypotenuse	The known and missing sides are adjacent and hypotenuse
Tangent	Opposite and adjacent	The known and missing sides are opposite and adjacent

Worked example

Estimate a tree’s height

A student stands

18

m from the base of a tree. The angle of elevation from the student’s eye level to the top of the tree is

34^\circ

. Estimate how far the top is above the student’s eye level. Round to the nearest tenth of a metre.

Sketch and label
Draw a right triangle. The horizontal distance is $18$ m, so it is adjacent to the $34^\circ$ angle. The height above eye level is opposite that angle. The line of sight is the hypotenuse, but it is not needed.
Choose the ratio
Tangent compares the opposite side with the adjacent side. It is the correct choice because those are the unknown and known lengths.
$\tan 34^\circ=\frac{h}{18}$
Solve for the height
Multiply both sides by $18$ to isolate $h$ . Evaluate the tangent in degree mode, then keep extra digits until rounding the final measurement.
$h=18\tan 34^\circ\approx 12.1\text{ m}$

Answer: The top of the tree is about

12.1

m above the student’s eye level.

Check: The height is less than the

18

m horizontal distance, which is reasonable for a

34^\circ

angle. The result is a length, so it is reported in metres. This does not include the student’s eye height above the ground.

Common mistakes and how to avoid them

Calling a side opposite even though it is beside the marked angle.

Correction: Mark the chosen angle first. Opposite is across from that angle; adjacent touches it and is not the hypotenuse.

Using the hypotenuse as the adjacent side.

Correction: The hypotenuse is always across from the right angle. The adjacent side is one of the two sides that form the marked acute angle.

Choosing a ratio because its name is familiar, without checking the sides.

Correction: Identify the known and unknown sides, then select the ratio that includes both.

Leaving off units or rounding too early.

Correction: Keep the calculator value during the steps, round the final measurement, and include the appropriate unit.

Lesson summary

A right triangle has one $90^\circ$ angle, and its longest side is the hypotenuse.
Opposite and adjacent are named from the chosen acute angle.
Use sine for opposite and hypotenuse, cosine for adjacent and hypotenuse, and tangent for opposite and adjacent.
Sketch the real situation, choose the matching ratio, solve for the missing length, and check the units and reasonableness.

Check your understanding

Question 1

A right triangle has a

40^\circ

angle. The adjacent side is

9

cm, and the opposite side is unknown. Which equation can be used to find the opposite side

x

?

$\tan 40^\circ=\frac{x}{9}$
$\sin 40^\circ=\frac{x}{9}$
$\cos 40^\circ=\frac{x}{9}$
$\tan 40^\circ=\frac{9}{x}$

Show answer and explanation

\tan 40^\circ=\frac{x}{9}

Tangent compares opposite with adjacent. Here,

x

is opposite and

9

cm is adjacent.

Question 2

A ladder is

5

m long and makes a

62^\circ

angle with the level ground. Which side is the hypotenuse?

The vertical height reached by the ladder
The ladder
The horizontal distance from the wall
The side opposite the right angle is not part of the triangle

Show answer and explanation

The ladder

The ladder is across from the right angle formed by the wall and ground, so it is the hypotenuse.

Key terms

Right triangle: A triangle with one angle measuring $90^\circ$ .
Hypotenuse: The side across from the right angle in a right triangle.
Opposite side: The side across from the chosen acute angle.
Adjacent side: The side beside the chosen acute angle that is not the hypotenuse.
Trigonometric ratio: A comparison of two side lengths in a right triangle, such as sine, cosine, or tangent.
Angle of elevation: An angle measured upward from a horizontal line to a line of sight.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MFM2P), study topic M6. It is a study resource, not an official curriculum publication.

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M6 · Solve real measurement problems with right-triangle trigonometry