Hint
Replace \(\displaystyle \sin ^{2}x\) by \(\displaystyle 1-\cos ^{2}x\). Check every algebraic cosine value against [−1,1] before taking an inverse.
Worked solution
(a)
Substitution gives \(\displaystyle u(2u-3)=1-u^{2}\). Expanding and collecting yields \(\displaystyle 3u^{2}-3u-1=0\).
Answer: \(3u^2-3u-1=0\).
(b)
The quadratic formula gives \(\displaystyle u= \frac{3\pm \sqrt{21}}{6}\). The plus value is approximately 1.26376>1, so cannot be a real cosine. The minus value is approximately −0.263763, inside [−1,1].
Answer: \(\displaystyle u= \frac{3\pm \sqrt{21}}{6}\); only \(\displaystyle \frac{3- \sqrt{21}}{6}\) is admissible.
(c)
For \(\displaystyle c= \frac{3- \sqrt{21}}{6}\), the principal angle is \(\displaystyle \arccos c\approx 1.837717212\). The second angle is \(\displaystyle 2\pi -\arccos c\approx 4.445468095\). Both satisfy the original equation and interval, giving 1.84 and 4.45 rad to 3 significant figures.
Answer: \(x=1.84,4.45\) rad.
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official examination-board mark scheme. Award stated criteria once each, up to the part maximum. Accept valid equivalent methods. Follow-through applies only where explicitly stated; do not award an accuracy criterion for an incorrect final result.
(a) · 3 marks
- 1 mark: Use \(\displaystyle \sin ^{2}x=1-\cos ^{2}x\).
- 1 mark: Expand and collect the quadratic terms.
- 1 mark: Obtain \(\displaystyle 3u^{2}-3u-1=0\).
Alternative methods: An equivalent correct rearrangement is accepted.
Common error: Replacing \(\displaystyle \sin ^{2}x\) by 1−cos x without the square.
(b) · 2 marks
- 1 mark: Give both exact quadratic roots.
- 1 mark: Reject the plus root using the cosine range and retain the minus root.
Alternative methods: Exact inequalities instead of decimal comparisons are accepted.
Common error: Applying arccos to a value larger than one.
(c) · 3 marks
- 1 mark: Obtain the principal radian angle from the exact admissible root.
- 1 mark: Find the second branch 2π minus that angle.
- 1 mark: Give both solutions correctly rounded to 3 significant figures with the original domain respected.
Alternative methods: Equivalent inverse-cosine notation followed by correct rounding is accepted.
Common error: Keeping only the principal inverse value or switching calculator units to degrees.
Total: 8 marks.