Sign error in the cosine rule H
"The Sign Slip"
The Mistake in Action
Find side $a$ when $b = 7$, $c = 9$, and $A = 120°$.
Wrong: $a^2 = 7^2 + 9^2 - 2(7)(9)\cos 120°$ $a^2 = 49 + 81 - 126(-0.5)$ $a^2 = 130 - (-63)$ $a^2 = 67$ $a = 8.2\text{cm}$
Why It Happens
Students correctly identify $\cos 120° = -0.5$ but then make an error when handling the double negative.
The Fix
When $\cos A$ is negative (obtuse angle), the formula becomes: $$a^2 = b^2 + c^2 - 2bc \times (\text{negative})$$ $$a^2 = b^2 + c^2 + \text{positive number}$$
Correct working: $a^2 = 49 + 81 - 126(-0.5)$ $a^2 = 130 + 63 = 193$ $a = 13.9\text{cm}$
This makes sense: an obtuse angle opposite side $a$ means $a$ is the longest side.
Spot the Mistake
Can you identify where this student went wrong?
$a^2 = 7^2 + 9^2 - 2(7)(9)\cos 120°$
$a^2 = 130 - (-63) = 67$
$a = 8.2\text{cm}$
Click on the line that contains the error.
Related Topics
Learn more about the underlying maths: