IB Maths AA SL AA 3.3 Trigonometric identities and equations questions

Exam-style questions in the shape of Paper 1 and Paper 2, with full markschemes. Read the markscheme, or type, photograph or draw your answer and get it marked.

Paper

Paper 1, no calculator Paper 2, calculator

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Question 1

Paper 1, no calculatorEasy4 marks

A point moves around a circle of radius 1. Its position is described by an angle $x$ measured from the positive horizontal axis, where $0\leq x\leq 2\pi$

At one measurement, the point’s vertical coordinate is $\frac12$ At another measurement, its horizontal coordinate is $-\frac{\sqrt3}{2}$

(a)

Determine all possible values of $x$ for the first measurement.

[2]

(b)

Determine all possible values of $x$ for the second measurement.

[2]

Question 2

Paper 2, calculatorMedium6 marks

For $0\leq x<2\pi$ let $f(x)=2\sin^2x+3\cos x-1$

(a)

Solve $f(x)=0$ analytically. Give all solutions in the interval to 3 significant figures.

[4]

(b)

Hence, use a GDC to check graphically that these are the only solutions in the stated interval.

[2]

Question 3

Paper 1, no calculatorMedium6 marks

Figure 1 shows a point $P$ on the unit circle. The angle $\theta$ is measured anticlockwise from the positive $x$ -axis to $OP$ and $\theta$ lies in quadrant II.

(a)

Determine $\cos\theta$ and $\tan\theta$

[3]

(b)

Hence, determine the exact values of $\sin 2\theta$ and $\cos 2\theta$

[3]

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