3.10.2 Cosine Rule

3.10.2 Cosine rule

The cosine rule (also known as the law of cosines) is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles.

• Specifically, the cosine rule states that:

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

where \( a, b \) and \( c \) are the lengths of the sides of the triangle, and \( C \) is the angle opposite the side of length \( c \).

• To use the cosine rule to find missing lengths and angles in a triangle, follow these general steps:

• Identify the side and angle you are looking for.

• Label them with the appropriate variable (e.g., \( c \) for the missing side, \( C \) for the missing angle).

• Identify the two known sides and the angle opposite the missing side or diagonal.

• Label them with the appropriate variables (e.g., \( a, b \) and \( A \) if you’re looking for side \( c \)).

• Substitute the known values into the cosine rule formula:

\( c^2=a^2+b^2-2 \ ab \ cos C \) or \( a^2=b^2+c^2-2 \ bc \ cos A \) or \( b^2=a^2+c^2-2 \ ac \ cos B \)

depending on which variable you are solving for.

• Solve the equation for the missing variable by using algebraic manipulation.

• If you are solving for an angle, use the inverse cosine function (\( cos^{-1} \) or arc \ cos) to find its value.

• Round your final answer to the appropriate number of significant figures.

Worked example:

In triangle \( ABC \), \( AB = 5 \) cm , \( AC = 7 \) cm, and \( BC = 6 \) cm. Find the measure of angle \( A \).

We are looking for angle \( A \), so we’ll label it with the variable \( A \).

We know the lengths of sides \( AB \), \( AC \), and \( BC \), so we’ll label them with \( a, b \) and \( c \), respectively. The angle opposite side \( AB \) is angle \( C \).

Using the cosine rule formula for angle \( A \), we get:

\( a^ 2 = b^2+c^2-2 \ bc \ cos(A) \)

Substituting the known values, we get

\( 5^2=7^2+6^2-2(7)(6) \ cos(A) \)

\( 25 = 85 – 84 \ cos(A) \)

\( 84 \ cos(A)=60 \)

\( cos(A) = \frac{60}{84} \)

Solving for \( A \) using the inverse cosine function, we get

\( A = cos^{-1}(\frac{60}{84}) \)

\( A \approx 34.5° \)

Therefore, the measure of angle \( A \) is approximately \( 34.5 \) degrees.

Tip:

Remember to check that your calculator is in degree mode.

Worked example:

Find the length of the side \( c \) in the triangle below, given that \( a=4 \) , \( b=7 \), and \( C=45° \)

To find the length of \( c \).

Applying the formula, we get

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

\( c^2=4^2+7^2-2(4)(7) \ cos(45) \)

\( c^2=16+49-56 \ cos(45) \)

\( c^2=65-39.6 \)

\( c^2=25.4 \)

Taking the square root of both sides, we get

\( c = \sqrt{25.4} \)

\( c \approx 5.04 \)

Therefore, the length of the side \( c \) is approximately \( 5.04 \) units.