The Cosine Rule (HSC SSCE Mathematics Standard): Revision Notes

Worked Example: Finding an Unknown Side

Let's find the value of $x$, correct to two decimal places, in a triangle where two sides measure $12$ and $6$, with an included angle of $76°$.

Step 1: Write the cosine formula to find a side.

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

Step 2: Substitute the values ($a = 12$, $b = 6$, $C = 76°$).

$x^2 = 12^2 + 6^2 - 2 \times 12 \times 6 \times \cos 76°$

Step 3: Calculate the value of $x^2$.

$x^2 = 144 + 36 - 144 \times \cos 76°$

$x^2 = 180 - 144 \times 0.2419...$

$x^2 = 180 - 34.836...$

$x^2 = 145.163247$

Step 4: Take the square root of both sides.

$x = \sqrt{145.163247}$

$x = 12.05 \text{ (correct to two decimal places)}$

Exam tip: Always check your calculator is in degree mode when working with angles in degrees!

Worked Example: Finding an Unknown Angle

Find the value of angle $\theta$ in a triangle with sides $8$, $11$, and $12$, where $\theta$ is opposite the side of length $8$. Answer in degrees, correct to one decimal place.

Step 1: Write the cosine formula to find an angle.

$\cos X = \frac{y^2 + z^2 - x^2}{2yz}$

Step 2: Substitute the values ($x = 8$, $y = 12$, $z = 11$, $X = \theta$).

$\cos \theta = \frac{12^2 + 11^2 - 8^2}{2 \times 12 \times 11}$

Step 3: Calculate the value of $\cos \theta$.

$\cos \theta = \frac{144 + 121 - 64}{264}$

$\cos \theta = \frac{201}{264}$

$\cos \theta = 0.7613636364$

Step 4: Use your calculator to find $\theta$.

$\theta = \cos^{-1}(0.7613636364)$

$\theta = 40.41543902°$

Step 5: Write the answer correct to one decimal place.

$\theta = 40.4°$

Exam tip: Remember to use the inverse cosine function ($\cos^{-1}$ or arccos) on your calculator to find the angle from the cosine value.

Worked Example: Real-World Application

Samuel is playing football and wants to shoot for goal. He is positioned $2.5$ m from one goalpost and $3.2$ m from the other goalpost. The goal is $2$ m wide. What is the size of angle $\theta$ that Samuel has to aim through to score? Answer correct to the nearest minute.

Step 1: Write the cosine formula to find an angle.

$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$

Step 2: Substitute the values ($a = 2$, $b = 2.5$, $c = 3.2$, $A = \theta$).

$\cos \theta = \frac{2.5^2 + 3.2^2 - 2^2}{2 \times 2.5 \times 3.2}$

Step 3: Calculate the value of $\cos \theta$.

$\cos \theta = \frac{6.25 + 10.24 - 4}{16}$

$\cos \theta = \frac{12.49}{16}$

$\cos \theta = 0.780625$

Step 4: Use your calculator to find $\theta$.

$\theta = \cos^{-1}(0.780625)$

$\theta = 38.68216452°$

Step 5: Convert decimal degrees to degrees and minutes.

$\theta = 38°41'$

(The decimal part $0.68216452° \times 60 = 40.93$ minutes $\approx 41$ minutes)

Step 6: Write the answer in words.

The size of the angle for Samuel to score a goal is $38°41'$.

Exam tip: To convert decimal degrees to minutes, multiply the decimal part by 60. For example, $0.682° \times 60 \approx 41$ minutes.