Hint
The number is nonzero. Read the positive polar radius.
Worked solution
(a)
Use the exact sine and cosine of the given angle.
Answer: \(-1 + \sqrt{3} i\)
(b)
Square the modulus and double the angle, then reduce the angle to the principal interval.
Answer: \(4\); \(\displaystyle - \frac{2 \pi}{3}\)
(c)
Double the angle and reduce into (−π,π].
\(\displaystyle - \frac{2}{3}\)
Answer: \(\displaystyle - \frac{2}{3}\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit complete valid alternatives and consequential reasoning; an unexplained final answer does not earn a method mark.
(a) · 2 marks
- 1 mark: Evaluate cosine and sine of the stated argument with the positive modulus.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not leave a principal argument outside its interval.
(b) · 2 marks
- 1 mark: Square the modulus, double the argument and reduce it into the stated principal interval.
- 1 mark: Give all results requested in this part, with the correct signs, exact values, units, branch and boundary inclusion where applicable.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: A modulus is nonnegative and is positive for a nonzero number.
(c) · 2 marks
- 1 mark: Double the angle and reduce into (−π,π].
- 1 mark: Give the correct requested result, retaining signs, units and any domain restrictions.
Alternative methods: Any complete mathematically valid alternative method with the same restrictions and equivalent exact result is acceptable.
Common error: Do not leave a principal argument outside its interval.
Total: 6 marks.