Hint
The reference angle here is acute. Use 2sinθcosθ.
Worked solution
(a)
Use 2sinθcosθ.
\(\displaystyle \frac{840}{841}\)
Answer: \(\displaystyle \frac{840}{841}\)
(b)
Subtract the squared sine from the squared cosine.
\(\displaystyle - \frac{41}{841}\)
Answer: \(\displaystyle - \frac{41}{841}\)
(c)
Use \(\displaystyle 1-\cos 2\theta =2\sin ^{2}\theta\) and \(\displaystyle \sin 2\theta =2\sin \theta \cos \theta\). Cancel only where the denominator is nonzero to obtain \(\displaystyle \frac{\sin \theta }{\cos \theta}\).
Answer: \(\displaystyle \frac{21}{20}\)
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: Use 2sinθcosθ.
- 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: cos2θ is not \(\displaystyle \cos ^{2}\theta\).
(b) · 2 marks
- 1 mark: Subtract the squared sine from the squared cosine.
- 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: A numerical example cannot prove a general identity.
(c) · 3 marks
- 1 mark: Transform the left side using double-angle identities on the common domain.
- 1 mark: Evaluate the resulting tangent using the given exact ratios.
- 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: cos2θ is not \(\displaystyle \cos ^{2}\theta\).
Total: 7 marks.