T8 Explore the development of the cosine law for acute triangles

Ontario Grade 10 Mathematics

Trigonometry

Use a perpendicular height, right triangles, and the Pythagorean theorem to develop a new side relationship.

The Pythagorean theorem connects the sides of a right triangle. But an acute triangle does not have a right angle, so we cannot apply that theorem directly to the whole triangle. We can still use it by drawing a perpendicular height inside the acute triangle. This creates right triangles. We will use their side lengths to develop the cosine law and see why the angle affects the length of the opposite side.

What you will learn

Review the Pythagorean theorem and cosine in a right triangle.
Describe how a perpendicular height divides an acute triangle into right triangles.
Explain how the height and a cosine-based side segment lead to the cosine law.
Use the cosine law to find a side opposite a known acute angle.

1. Bridge: right triangles and cosine

A right triangle has one angle measuring

90^\circ

. The hypotenuse is the side opposite that angle. The Pythagorean theorem says that the square of the hypotenuse equals the sum of the squares of the other two sides.

Cosine describes a side relationship in a right triangle. For an acute angle in that triangle, cosine is the length of the adjacent side divided by the length of the hypotenuse. The adjacent side is the side next to the chosen angle that is not the hypotenuse. For example, if the adjacent side is

4

and the hypotenuse is

5

, the cosine of the angle is

\frac{4}{5}

.

An acute angle measures less than

90^\circ

. In this lesson, we use cosine for acute angles only. A perpendicular height is a line segment that meets a side at

90^\circ

. Drawing one can create right triangles inside a larger acute triangle.

z^2=x^2+y^2

The Pythagorean theorem applies to right triangles.
For an acute angle in a right triangle, cosine is adjacent side divided by hypotenuse.
A perpendicular height creates right triangles inside the acute triangle.

2. Build right triangles inside an acute triangle

Imagine an acute triangle with angle

C

between sides

a

and

b

. Name the side opposite angle

C

as

c

. The side names help us keep track of which parts of the triangle meet at the angle and which side is opposite it.

Draw a perpendicular height from the vertex opposite side

a

to side

a

. Since the triangle is acute, the foot of the height lies on side

a

, between its endpoints. The height divides the original triangle into two right triangles.

In the right triangle beside angle

C

, side

b

is the hypotenuse. The horizontal part of that side along side

a

is adjacent to angle

C

. By the definition of cosine, this part has length

b\cos C

. Call the height

h

. Using the Pythagorean theorem in this right triangle gives a relationship for

h^2

.

In the other right triangle, the horizontal leg is the remaining part of side

a

, so its length is

a-b\cos C

. Its hypotenuse is

c

, and its other leg is the same height,

h

. Apply the Pythagorean theorem again. Then substitute the first relationship for

h^2

. The height cancels, leaving a rule that uses only the side lengths and angle

C

.

Here is the algebra in one line. Expanding the square creates a term involving

b^2\cos^2 C

. That term cancels with the matching term from the first right triangle. The remaining expression is the cosine law for this triangle.

c^2=(a-b\cos C)^2+b^2-(b a? )

The segment along side $a$ under side $b$ has length $b\cos C$ .
The remaining part of side $a$ has length $a-b\cos C$ .
The same height appears in both right triangles, so it can be removed from the final relationship.

3. Read and use the cosine law

The developed rule relates the two sides that meet at angle

C

to the side opposite that angle. The angle matters because it determines the length of the segment

b\cos C

along side

a

.

For an acute angle, cosine is positive. Therefore, the adjustment term in the cosine law is positive before it is subtracted. The squared opposite side is less than the sum of the squares of the two sides meeting at the angle. This helps explain how the angle changes the side relationship.

To use the rule, first identify the angle and the side opposite it. Then identify the two sides that meet at that angle. Only those two sides belong in the adjustment term. After finding the square of the unknown side, take its positive square root to find the side length.

c^2=a^2+b^2-2ab\cos C

The angle must be between the two sides in the adjustment term.
The left side of the rule is the square of the side opposite the angle.
For an acute angle, the adjustment is subtracted.

4. Guided example and independent practice

In the example, the two given sides meet at the given acute angle. The opposite side is unknown. We substitute into the cosine law, calculate its square, and take the positive square root.

Independent practice: In an acute triangle, sides of lengths

8

and

6

meet at

60^\circ

. Set up the cosine-law expression for the square of the opposite side, evaluate it, and compare it with the sum of the two side squares. The comparison should reflect that the cosine adjustment is subtracted.

c=\sqrt{a^2+b^2-2ab\cos C}

Match the included angle to the two sides that form it.
Substitute carefully before calculating.
A side length is positive, so use the positive square root.

Pieces created by the perpendicular height

Part	Length	Why
Whole base	$a$	It is opposite angle $C$ .
Segment along the base under side $b$	$b\cos C$	Cosine is adjacent side divided by hypotenuse.
Remaining base segment	$a-b\cos C$	Subtract the first segment from the whole base.
Other right-triangle leg	$h$	It is the same perpendicular height in both right triangles.

Worked example

Find the side opposite a known acute angle

In an acute triangle, sides

a=7

and

b=5

meet at angle

C=60^\circ

. Find the length of the opposite side

c

. Use

\cos 60^\circ=0.5

.

Match the parts
Angle $C$ lies between sides $a$ and $b$ . The side opposite it is $c$ , so this is the matching form of the cosine law.
$c^2=a^2+b^2-2ab\cos C$
Substitute the values
Replace the side lengths and cosine value with the information given. Keep the multiplication together so the subtraction is clear.
$c^2=7^2+5^2-2(7)(5)(0.5)$
Calculate the side length
The calculation gives the square of the unknown side. Since a length is positive, take the positive square root.
$c^2=39\qquad c=\sqrt{39}\approx6.24$

Answer: The opposite side is approximately

6.24

units long.

Check: The squared result,

39

, is less than

7^2+5^2=74

. This is consistent with subtracting the positive cosine adjustment for an acute angle.

Common mistakes and how to avoid them

Using an angle that is not between the two known sides.

Correction: The angle in the adjustment term is formed by those two sides. Check which sides meet at the angle before substituting.

Adding the adjustment for an acute angle.

Correction: Cosine is positive for an acute angle, so the adjustment term is subtracted.

Stopping after finding the square of the unknown side.

Correction: Take the positive square root to find the side length.

Applying the Pythagorean theorem to the whole acute triangle.

Correction: The whole triangle is not a right triangle. Apply the theorem to the two right triangles made by the height.

Lesson summary

Cosine in a right triangle is adjacent side divided by hypotenuse.
A perpendicular height divides an acute triangle into right triangles.
The height and the segment $b\cos C$ lead to the cosine law.
For angle $C$ , sides $a$ and $b$ meet at the angle, and $c$ is opposite it.
For an acute angle, the cosine adjustment is subtracted.

Check your understanding

Question 1

Sides

p

and

q

meet at acute angle

R

. Which side is opposite angle

R

in the matching cosine-law form?

The side called $r$
Side $p$
Side $q$
Either side $p$ or side $q$

Show answer and explanation

The side called

r

The two sides in the adjustment term meet at the angle. The remaining side,

r

, is opposite that angle.

Question 2

Sides

a=6

and

b=4

meet at

60^\circ

. What is the square of the opposite side?

$28$
$52$
$76$
$100$

Show answer and explanation

28

Substitute into the cosine law:

6^2+4^2-2(6)(4)(0.5)=36+16-24=28

.

Question 3

Why is the cosine adjustment subtracted for an acute angle?

Cosine is positive for an acute angle.
Cosine is negative for an acute angle.
The opposite side is always the longest side.
The height always equals one of the other sides.

Show answer and explanation

Cosine is positive for an acute angle.

Cosine is positive for an acute angle. The positive adjustment is therefore subtracted in the cosine law.

Key terms

Acute angle: An angle with a measure less than $90^\circ$ .
Adjacent side: In a right triangle, the side next to a chosen acute angle that is not the hypotenuse.
Cosine: For an acute angle in a right triangle, the ratio of the adjacent side length to the hypotenuse length.
Hypotenuse: The side opposite the right angle in a right triangle.
Perpendicular: Meeting at a right angle.
Cosine law: A rule relating the sides of a triangle to the cosine of one of its angles.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic T8. It is a study resource, not an official curriculum publication.

Official curriculum reference

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T8 · Explore the development of the cosine law for acute triangles

Ontario Grade 10 Mathematics

Use a perpendicular height, right triangles, and the Pythagorean theorem to develop a new side relationship.

What you will learn

1. Bridge: right triangles and cosine

2. Build right triangles inside an acute triangle

3. Read and use the cosine law

4. Guided example and independent practice

Pieces created by the perpendicular height

Find the side opposite a known acute angle

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MPM2D

About this lesson