Solve problems involving the cosine rule

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Cosine Rule

9.2

Cosine Rule

Definition of Cosine Rule

Cosine Rule Formulas

Triangle illustration featuring sides labeled a, b, c and angles marked A, B, and C.

For any triangle \(ABC\):

\(a^2=b^2+c^2-2ab\cos{A}\)

\(b^2=a^2+c^2-2ab\cos{B}\)

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

Where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite those sides.

Conditions for Using the Cosine Rule

When to Use

When two sides and the included angle are known.
When all three sides are known, and you want to find an angle.

Not Suitable For

Right-angled triangles, where the Pythagorean theorem and basic trigonometric ratios can be used instead.

Applications of the Cosine Rule

Finding a Side

When two sides and the included angle are known, use the cosine rule to find the unknown side.

Finding an Angle

When all three sides are known, use the cosine rule to find one of the angles.

Extension of Pythagorean Theorem

The cosine rule is a generalized form of the Pythagorean theorem that works for all types of triangles, not just right-angled ones.

Example

Question

Illustration of a triangle featuring sides AB at 25 cm, BC at 23 cm, and an angle B of 40 degrees.

In the diagram above, \(ABC\) is a scalene triangle.

Find the length of the \(AC\).

Solution

From diagram above, given:

\(AB=25\) cm,
\(BC=23\) cm,
\(\angle{B}=40^\circ\).

Apply the cosine rule to find the length of \(AC\):

\(\begin{aligned} AC^2&=AB^2+BC^2-2(AB)(BC) \cos 40^\circ \\\\ &=25^2+23^2-2(25)(23) \cos 40^\circ \\\\ &=273.05. \\\\ AC&=16.52 \text{ cm}. \end{aligned}\)

Cosine Rule

9.2

Cosine Rule

Definition of Cosine Rule

Cosine Rule Formulas

Triangle illustration featuring sides labeled a, b, c and angles marked A, B, and C.

For any triangle \(ABC\):

\(a^2=b^2+c^2-2ab\cos{A}\)

\(b^2=a^2+c^2-2ab\cos{B}\)

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

Where \(a\), \(b\), and \(c\) are the sides of the triangle, and \(A\), \(B\), and \(C\) are the angles opposite those sides.

Conditions for Using the Cosine Rule

When to Use

When two sides and the included angle are known.
When all three sides are known, and you want to find an angle.

Not Suitable For

Right-angled triangles, where the Pythagorean theorem and basic trigonometric ratios can be used instead.

Applications of the Cosine Rule

Finding a Side

When two sides and the included angle are known, use the cosine rule to find the unknown side.

Finding an Angle

When all three sides are known, use the cosine rule to find one of the angles.

Extension of Pythagorean Theorem

The cosine rule is a generalized form of the Pythagorean theorem that works for all types of triangles, not just right-angled ones.

Example

Question

Illustration of a triangle featuring sides AB at 25 cm, BC at 23 cm, and an angle B of 40 degrees.

In the diagram above, \(ABC\) is a scalene triangle.

Find the length of the \(AC\).

Solution

From diagram above, given:

\(AB=25\) cm,
\(BC=23\) cm,
\(\angle{B}=40^\circ\).

Apply the cosine rule to find the length of \(AC\):

\(\begin{aligned} AC^2&=AB^2+BC^2-2(AB)(BC) \cos 40^\circ \\\\ &=25^2+23^2-2(25)(23) \cos 40^\circ \\\\ &=273.05. \\\\ AC&=16.52 \text{ cm}. \end{aligned}\)

Cosine Rule

Cosine Rule

Chapter : Solution of Triangles

Topic : Solve problems involving the cosine rule

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