T6 Apply right-triangle trigonometry to real situations

Ontario Grade 10 Mathematics

Trigonometry

Build a triangle model, choose a ratio, and interpret your answer

A right triangle has one angle measuring

90^\circ

. You may already know how to find a missing side when two side lengths are given. Right-triangle trigonometry gives another way to connect an angle with side lengths. This is useful when a height or distance is difficult to measure directly. You will model a situation with a right triangle, choose a ratio, and explain what your result means.

What you will learn

Identify the opposite, adjacent, and hypotenuse sides in a right triangle.
Choose sine, cosine, or tangent to connect a known side and an unknown side.
Use a calculator in degree mode to solve a real situation.
State an answer with suitable units and check whether it is reasonable.

1. From a real situation to a triangle

Recall that perpendicular lines meet at a right angle. A ladder leaning against a wall can form a right triangle: the wall and ground meet at the right angle, and the ladder is the sloping side.

The two angles other than the right angle are acute angles. An acute angle is less than

90^\circ

. For example, the angle between level ground and a line of sight to the top of a tree is an acute angle.

A diagram is a model: a simplified picture that keeps the important parts of a situation. First decide what forms the right angle, which lengths are known, and which length you need. Mark the right angle before naming the sides.

Suppose someone stands on level ground and looks up at a tree. The ground distance and the height above the person's eyes can be drawn as the two sides that meet at the right angle. The line of sight to the top is the sloping side.

A useful model matches the important features of the real situation.
The hypotenuse is opposite the right angle. It is the longest side.
The names opposite and adjacent depend on the acute angle you select.

2. Name sides and choose a ratio

Choose the acute angle you will use. In this lesson, the symbol

\theta

(theta) represents that selected acute angle. Naming it first helps you identify the sides consistently.

The opposite side is directly across from

\theta

. The adjacent side touches

\theta

and is not the hypotenuse. The hypotenuse is across from the right angle, so its name does not change when you select a different acute angle.

A trigonometric ratio compares two side lengths in a right triangle and links them to an acute angle. Sine connects opposite and hypotenuse. Cosine connects adjacent and hypotenuse. Tangent connects opposite and adjacent. The memory aid “SOH-CAH-TOA” summarizes these pairings.

Choose the ratio that contains both the side you know and the side you need. If you know the adjacent side and need the opposite side, tangent is the matching ratio. Check the side names on your diagram before writing an equation.

A calculator can evaluate sine, cosine, or tangent. When an angle is given in degrees, set the calculator to degree mode. Keep enough digits during the calculation, then round the final measurement sensibly.

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

Sine connects opposite and hypotenuse.
Cosine connects adjacent and hypotenuse.
Tangent connects opposite and adjacent.
The selected acute angle determines which sides are opposite and adjacent.

3. Set up and solve

Use the situation and your diagram to identify the sides. When an angle is measured upward from level ground, the horizontal distance is adjacent to the angle, and the vertical rise is opposite. A ladder or line of sight may be the hypotenuse.

Write an equation before using the calculator. Substitute the angle and known side into the ratio that matches the unknown. Then rearrange the equation to isolate the unknown side. This connects each calculation to the triangle.

For example, if tangent relates the unknown height

h

to a known horizontal distance

d

, the equation has the form

\tan\theta=\frac{h}{d}

. Multiplying both sides by

d

gives

h=d\tan\theta

.

A calculator result is usually an approximation. Include the unit, such as metres, and say what the measurement represents. A height above eye level is not the same as a tree’s full height from the ground.

Check the result. A length cannot be negative. The hypotenuse must be longer than either other side. Also ask whether the size makes sense in the real situation.

Draw and label before calculating.
Use a ratio that contains the known side and the unknown side.
Keep units and explain what the measurement represents.
Round after calculating and check that the result is sensible.

4. Guided example and independent practice

Work through the example by following the diagram, side names, ratio, calculation, and interpretation. The angle of elevation is measured upward from a horizontal direction. In this problem, the answer is a vertical rise above eye level, not the tree’s full height.

For independent practice, draw and label a right-triangle model before calculating. A ramp rises

1.2

m over a horizontal distance of

3.5

m. Identify the acute angle at the ground and decide which ratio would connect the rise and horizontal distance. Then explain why your choice fits. Do not assume that every real situation gives enough information to calculate a numerical answer.

A well-labelled model makes the ratio easier to choose.
The answer must describe the quantity asked for, not a related measurement.

Side names and ratio choices

Ratio	Sides connected	Use when the known and unknown sides are
Sine	Opposite and hypotenuse	Opposite and hypotenuse
Cosine	Adjacent and hypotenuse	Adjacent and hypotenuse
Tangent	Opposite and adjacent	Opposite and adjacent

Worked example

Estimate a tree’s height above eye level

A student stands

24

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

38^\circ

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

Model the situation
Imagine a right triangle with the ground as the horizontal side and the vertical rise from eye level to the treetop as the vertical side. The angle of elevation is measured upward from the horizontal direction. The $24$ m ground distance is adjacent to the $38^\circ$ angle. The height we want is opposite.
Choose tangent
The known side is adjacent and the unknown side is opposite. Tangent connects those two sides, so write an equation using the selected angle.
$\tan 38^\circ=\frac{h}{24}$
Calculate and interpret
Multiply both sides by $24$ to isolate the height. Use degree mode to evaluate the tangent. The result is the vertical distance from eye level to the top, not the tree’s full height.
$h=24\tan 38^\circ\approx18.8\text{ m}$

Answer: The top of the tree is approximately

18.8

m above the student’s eye level.

Check: The height is positive. The sloping line of sight would be longer than either leg of the triangle. A rise of about

19

m across a

24

m horizontal distance is reasonable for an angle of

38^\circ

.

Common mistakes and how to avoid them

Calling the same side opposite in every diagram.

Correction: Opposite and adjacent are named relative to the selected acute angle. Mark that angle first.

Choosing the hypotenuse by appearance alone.

Correction: The hypotenuse is the side opposite the right angle.

Using a ratio with the wrong pair of sides.

Correction: Label the known and unknown sides, then select the ratio containing both.

Using degree values while the calculator is in another angle setting.

Correction: Check that the calculator is in degree mode before evaluating a ratio.

Giving a number without a unit or explanation.

Correction: Include the unit and say what the measurement represents, such as height above eye level.

Lesson summary

Draw a right-triangle model and mark the right angle.
Select an acute angle, then identify opposite, adjacent, and hypotenuse.
Match the known and unknown sides with sine, cosine, or tangent.
Solve, include units, round sensibly, and check whether the result fits the situation.

Check your understanding

Question 1

A right triangle has a

31^\circ

angle. The adjacent side is

12

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

x

?

$\tan 31^\circ=\frac{x}{12}$
$\sin 31^\circ=\frac{x}{12}$
$\cos 31^\circ=\frac{x}{12}$
$\tan 31^\circ=\frac{12}{x}$

Show answer and explanation

\tan 31^\circ=\frac{x}{12}

Tangent connects the opposite and adjacent sides. Here,

x

is opposite and

12

cm is adjacent.

Question 2

A student calculates a height using an angle of elevation given in degrees. What calculator setting should the student check?

Degree mode
A setting that changes metres to centimetres
A setting that changes the triangle’s right angle
No setting is needed for trigonometric ratios

Show answer and explanation

Degree mode

The angle is measured in degrees, so the calculator should evaluate the trigonometric ratio in degree mode.

Key terms

Right triangle: A triangle with one angle measuring $90^\circ$ .
Acute angle: An angle greater than $0^\circ$ and less than $90^\circ$ .
Hypotenuse: The side opposite the right angle in a right triangle.
Opposite side: The side directly across from the selected acute angle.
Adjacent side: A side that touches the selected acute angle and is not the hypotenuse.
Angle of elevation: An angle measured upward from a horizontal direction to a line of sight.
Trigonometric ratio: A comparison of two side lengths in a right triangle, linked to an acute angle.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic T6. It is a study resource, not an official curriculum publication.

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T6 · Apply right-triangle trigonometry to real situations