2023HSCShort answer

Q11 (e)

3 marks

Question

Solve $\cos\theta+\sin\theta=1$ for $0\leq\theta\leq2\pi$ .

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	3
Obtains an equation or graph involving a single trigonometric function OR obtains a quadratic equation in $t$ , and attempts to solve it OR equivalent merit	2
Uses the $t$ -substitution, attempts to write $\cos\theta+\sin\theta$ as a single trigonometric function, or equivalent merit	1

Sample answer:

\text{Let }t=\tan\frac{\theta}{2}

\frac{1-t^2}{1+t^2}+\frac{2t}{1+t^2}=1

1-t^2+2t=1+t^2

2t^2-2t=0

2t(t-1)=0

\therefore\quad t=0\quad\text{or}\quad t=1

\tan\frac{\theta}{2}=0\quad\text{or}\quad\tan\frac{\theta}{2}=1

\frac{\theta}{2}=0,\pi,2\pi,\ldots\quad\text{or}\quad\frac{\theta}{2}=\frac{\pi}{4},\frac{5\pi}{4},\ldots

\theta=0,2\pi\quad\text{or}\quad\theta=\frac{\pi}{2}

\therefore\quad\theta=0,\frac{\pi}{2},2\pi

Supplied marking guidance · human reviewed

Compare this with your working, not only your final sentence.

Syllabus alignment

Trigonometric Functions

Trigonometric EquationsUse identities and general solutions to solve trigonometric equations and interpret solutions in context.

Spot a problem? Send feedback