Edexcel A-Level Mathematics AS P2.6 Trigonometry Questions

Practise rewriting trigonometric expressions with identities, then solving equations over stated degree or radian intervals with exact or rounded answers.

Exam points

find all solutions in a stated interval, giving degrees or radians to the required accuracy

Covered syllabus points

P2.6.1 - Core trigonometric identities P2.6.2 - Solving trigonometric equations

[Maximum number: 8]

In this question you must show detailed reasoning.

Solutions relying entirely on calculator technology are not acceptable.

(a)

Show that the equation

(3 \cos \theta-\tan \theta) \cos \theta=2

can be written as

3 \sin ^{2} \theta+\sin \theta-1=0

[ 3 ]

$\cos \theta(3 \cos \theta-\tan \theta)=2 \Rightarrow \cos \theta\left(3 \cos \theta-\frac{\sin \theta}{\cos \theta}\right)=2$

or e.g.
$\cos \theta(3 \cos \theta-\tan \theta)=2 \Rightarrow 3 \cos ^{2} \theta-\sin \theta=2$

or e.g.
$\cos \theta(3 \cos \theta-\tan \theta)=2 \Rightarrow 3 \cos ^{2} \theta-\frac{\sin \theta}{\cos \theta} \cos \theta=2$ $3\left(1-\sin ^{2} \theta\right)-\sin \theta=2$ $3 \sin ^{2} \theta+\sin \theta-1=0 *$

A1*

(3)

(b)

Hence solve for $-\frac{\pi}{2} \leqslant x \leqslant \frac{\pi}{2}$

(3 \cos 2 x-\tan 2 x) \cos 2 x=2

[ 5 ]

Solving the quadratic from part (a) gives
$\sin 2x=\frac{-1\pm\sqrt{13}}{6}$
or approximately 0.434 and -0.768.
Then solve $2x=\sin^{-1}(0.434)$ or $2x=\sin^{-1}(-0.768)$ in the range $-\frac{\pi}{2}\le x\le\frac{\pi}{2}$ .
x=-1.13, -0.438, 0.225, 1.35.

[Maximum number: 4]

Figure 5

Figure 5 shows a sketch of part of the curve C with equation $y=\sin \left(\frac{x}{12}\right)$ , where x is measured in radians. The point M shown in Figure 5 is a minimum point on C.

(a)

the negative solution of the equation $\sin \left(\frac{x}{12}\right)=k$ that is closest to zero,

[ 2 ]

$-12 \pi-\alpha$

B1 ft

(b)

the smallest positive solution of the equation (x/12)=k.

[ 2 ]

$6\pi-\alpha$

B1 ft

All question bank results loaded

Covered syllabus points

Question 1

Question (a)

Question (b)

Question 2

Question (a)

Question (b)