Question 6
Solve the equation $\cos(\theta)=-0.6$ for $0^\circ\le\theta\le360^\circ$
Not quite. These two angles share a sine, not a cosine. $\cos 53.13^\circ=0.6$, which is positive, so $53.13^\circ$ can’t be a solution. In 2025, 10.9% of Queensland students chose (A).
Not quite. $53.13^\circ$ is the reference angle, and $\cos 53.13^\circ=+0.6$. Cosine is negative only in quadrants 2 and 3, so the first solution is $180^\circ-53.13^\circ=126.87^\circ$. In 2025, 7.7% of Queensland students chose (B).
Close. $126.87^\circ$ is right, but adding $180^\circ$ doesn’t keep the cosine. That lands in quadrant 4, where $\cos 306.87^\circ=+0.6$. The second solution is $360^\circ-126.87^\circ=233.13^\circ$. In 2025, 9.5% of Queensland students chose (C).
That’s it. Step through the hints below to see why. In 2025, 71.4% of Queensland students chose (D).
This is a solution. The point sits on the line $x=-0.6$, so $\cos\theta=-0.6$. Its mirror image in the $x$-axis, the grey point, is the other solution.
This is the second solution, $360^\circ-126.87^\circ$. It is the mirror image of $126.87^\circ$ in the $x$-axis, so it has the same cosine.
This is the reference angle, about $53.13^\circ$. Its cosine is $+0.6$, not $-0.6$, so it isn’t a solution.
Very close. Here $\cos\theta$ is within 0.005 of $-0.6$. Use the buttons below for the exact angles.
The point is on or to the right of the $y$-axis, so $\cos\theta\ge0$. Nothing in quadrants 1 or 4 can give $-0.6$.
The point is to the left of the $y$-axis, so $\cos\theta$ is negative. Both solutions are in quadrants 2 and 3, where the circle meets the dashed line.
Ignore the sign for a moment and find the reference angle. That is the acute angle each solution makes with the $x$-axis.
$$\cos^{-1}(0.6)=53.13^\circ$$
On the unit circle, $\cos\theta$ is the $x$-coordinate. It is negative to the left of the $y$-axis, in quadrants 2 and 3. So one solution sits in each of those quadrants.
Write the reference angle into each quadrant.
$\text{quadrant 2:}$$180^\circ-53.13^\circ=126.87^\circ$
$\text{quadrant 3:}$$180^\circ+53.13^\circ=233.13^\circ$
Both are inside $0^\circ\le\theta\le360^\circ$. Your calculator gives $\cos^{-1}(-0.6)=126.87^\circ$ straight away, and $360^\circ-126.87^\circ$ gives the other.
$$\theta=\cos^{-1}(-0.6)=126.87^\circ$$
$$\theta=360^\circ-126.87^\circ=233.13^\circ$$
Cosine is negative in quadrants 2 and 3, so these are the only two solutions in $0^\circ\le\theta\le360^\circ$.
How Queensland students answered in 2025
From the QCAA Mathematical Methods subject report for 2025. The percentages can add to less than 100 because some students didn’t answer.
Only mark this done when you could do it without help. Reading the solution does not count.
Question wording is from the 2025 QCAA Mathematical Methods external assessment, © State of Queensland (QCAA) 2025, licensed under CC BY 4.0, and has been adapted. Tangent Tuition is not affiliated with the QCAA.
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