Solving a Cosine Equation in Radians

Solving in radians uses the same quadrant logic as degrees, with $\pi$ in place of $180^\circ$. For $\cos x = -0.5$, cosine is negative in the second and third quadrants, and the answers come out as neat multiples of $\pi$.

Solve $\cos x = -0.5$ for $0 \le x \le 2\pi$. [4]

The working

Step 1, find the related acute angle. Ignoring the sign, $\cos^{-1}(0.5) = \dfrac{\pi}{3}$ (a standard exact value). (M1)

Step 2, place the solutions. Cosine is negative in the second and third quadrants:

• Second quadrant: $x = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3}$
• Third quadrant: $x = \pi + \dfrac{\pi}{3} = \dfrac{4\pi}{3}$

(M1, A1)

Step 3, check the interval. Both $\dfrac{2\pi}{3}$ and $\dfrac{4\pi}{3}$ lie in $[0, 2\pi]$: $x = \frac{2\pi}{3}, \ \frac{4\pi}{3} \quad \text{(A1)}$

Where the marks are won and lost

• Cosine is negative in the 2nd and 3rd quadrants. Getting the quadrants right is what places the solutions correctly; using the wrong ones gives wrong angles.
• The related acute angle is $\frac{\pi}{3}$ (from $\cos\frac{\pi}{3} = 0.5$), a value worth knowing exactly for radian work.
• Keep answers as exact multiples of $\pi$ where possible, and confirm they fall in $[0, 2\pi]$.

Common mistakes

• Using the first and fourth quadrants (where cosine is positive) by mistake.
• Giving decimals ($2.09, 4.19$) when exact multiples of $\pi$ are cleaner and expected.
• Missing one of the two solutions.

Full method: Solving Trig Equations notes. See also Exact Values & the Unit Circle. Topic home: Trigonometry pillar.