CAIE A-Level Mathematics 1.5.5 Trigonometric Equations

$\cos ^{2} \theta+2 \sin \theta \cos \theta+\sin ^{2} \theta=1$ leading to $2 \sin \theta \cos \theta=0$ or $\sin 2 \theta=0$

*B1

Or arriving at $\cos \theta=0$ or $\sin \theta=0$ or $\tan \theta=0$ after first
expanding and www.

$[\theta=] \quad 0, \frac{\pi}{2}, \pi$

DB
2,1,0

B2 for three correct answers only.
B1 for two correct answers and one incorrect or 3 correct answers plus other values in the range.
SC DB1 for correct 3 answers in degrees and no others.

Marking guidance:

Ignore extras outside of the range and allow decimal
equivalents.

3

Verifying 3 answers rather than expanding and solving 0/3.

$\cos 0+\sin 0=[1+0=] 1$ and $\cos \frac{\pi}{2}+\sin \frac{\pi}{2}[=0+1]=1$

B1

Checking both correct values. Do not allow solving an equation.
Condone use of 90 degrees.

$\cos \pi+\sin \pi[=-1+0]=-1$ or $\neq 1$

B1

WWW

$$\begin{aligned} \frac{\cos \theta+\sin \theta-1}{1-2 \sin ^{2} \theta}=2(\cos \theta+\sin \theta-1) \text { leading to } 1=2\left(1-2 \sin ^{2} \theta\right) \end{aligned}$$

*M1

Replacing LHS with the expression from (b) and attempting to simplify i.e. condone omission of
$(\cos \theta+\sin \theta-1)=0$ at this stage.
M0 for $0=2\left(1-2 \sin ^{2} \theta\right)$$k \sin ^{2} \theta=1$ or 3 leading to $\sin \theta=[ \pm] \sqrt{\frac{1 \text { or } 3}{k}}$$$\left[4 \sin ^{2} \theta=1 \text { leading to } \sin \theta= \pm \frac{1}{2}\right]$$

DM1

Dividing by k and taking the square root of a positive value <1.

This mark can be implied by the solutions $\frac{1}{6} \pi, \frac{5}{6} \pi$.

Solutions $0, \frac{1}{6} \pi, \frac{1}{2} \pi, \frac{5}{6} \pi$

A1

Allow 0, 0.524, 1.57, 2.62 AWRT.

If M0 SCB1 for $(\cos \theta+\sin \theta-1)=0 \Rightarrow 0, \frac{1}{2} \pi$.

If M0 SCB1 for all four correct answers and no others. Ignore answers outside of the range.

Answers in degrees A0.