CAIE A-Level Mathematics 2.3.2 Trig identities

Practise choosing trig identities to simplify expressions, prove results and solve equations within stated angle intervals.

[Maximum number: 8]

The expression $\mathrm{f}(\theta)$ is defined by $\mathrm{f}(\theta)=12 \sin \theta \cos \theta+16 \cos ^{2} \theta$ .

(a)

Express $\mathrm{f}(\theta)$ in the form $R \cos (2 \theta-\alpha)+k$ , where $R>0,0<\alpha<\frac{1}{2} \pi$ and k is a constant. State the values of R and k, and give the value of $\alpha$ correct to 4 significant figures.

[ 5 ]

Use correct identity for $\sin 2 \theta$ or $\cos 2 \theta$ (or both)

M1

Obtain $6 \sin 2 \theta+8 \cos 2 \theta(+8)$

A1

State R=10

B1 FT

FT their $a \sin 2 \theta+b \cos 2 \theta$ form.

Use appropriate trigonometry to find $\alpha$ using their $a \sin 2 \theta+b \cos 2 \theta$ form

M1

Marking guidance:

Allow 0.927.

Obtain $10 \cos (2 \theta-0.6435)+8$

A1

(b)

Find the smallest positive value of $\theta$ satisfying the equation $\mathrm{f}(\theta)=17$ .

[ 3 ]

State or imply $\cos (2 \theta-0.6435)=0.9$

B1 FT

FT their $R, \alpha$ and k provided RHS less than 1 .

Carry out correct process to find any positive value of $\theta$

M1

0.547

Obtain 0.0962

A1

AWRT

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2.3.2—Trig identities question 1

Question (a)

Question (b)