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Solving Trigonometric Equations

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen, paper and a calculator handy, to work out your answers.

Question 12 marks

Solve 4\tan x + 7 = 0 for 0^\circ \leqslant x \leqslant 360^\circ

Give your answers to 1 decimal place.

Give every value, separated by commas

Hint

Your calculator gives a negative angle; the tan graph repeats every 180^\circ, so keep adding 180^\circ.

Worked solution
  1. \tan x=-\dfrac74=-1.75
  2. \tan^{-1}(-1.75)=-60.3^\circ, which is outside the range
  3. tan repeats every 180^\circ
  4. -60.3^\circ+180^\circ=119.7^\circ
  5. 119.7^\circ+180^\circ=299.7^\circ
  6. x=119.7^\circ and x=299.7^\circ

Question 21 mark

One solution of \cos x = -0.7 is x = 134.4^\circ, to 1 decimal place.

Which of these is the other solution for 0^\circ \leqslant x \leqslant 360^\circ?

Hint

The graph of y=\cos x for 0^\circ to 360^\circ is symmetrical about x=180^\circ.

Worked solution
  1. The cos graph is symmetrical about x=180^\circ
  2. So the other solution is 360^\circ-134.4^\circ
  3. =225.6^\circ
  4. (45.6^\circ uses the sine symmetry 180^\circ-x, 314.4^\circ adds 180^\circ, which is the tan rule, and -134.4^\circ is outside the range.)

Question 3Challenge5 marks

Solve 10\cos^2 x - \sin x = 8 for 0^\circ \leqslant x \leqslant 360^\circ

(a)

Use \sin^2 x + \cos^2 x = 1 to write the equation as a quadratic in \sin x

Work out the two possible values of \sin x

3 marks

Give every value, separated by commas

(b)

Hence solve 10\cos^2 x - \sin x = 8 for 0^\circ \leqslant x \leqslant 360^\circ

Give your answers to 1 decimal place where appropriate.

2 marks

Give every value, separated by commas

Hint

Use \sin^2 x+\cos^2 x=1 to replace \cos^2 x, so the equation only contains \sin x; then treat it as a quadratic.

Worked solution

Part (a)

  1. Replace \cos^2 x with 1-\sin^2 x
  2. 10(1-\sin^2 x)-\sin x=8
  3. 10-10\sin^2 x-\sin x-8=0
  4. 10\sin^2 x+\sin x-2=0
  5. Factorise: (2\sin x+1)(5\sin x-2)=0
  6. \sin x=-\dfrac12 or \sin x=\dfrac25

Part (b)

  1. \sin x=\dfrac25: \sin^{-1}0.4=23.6^\circ
  2. and 180^\circ-23.6^\circ=156.4^\circ
  3. \sin x=-\dfrac12: \sin^{-1}(-0.5)=-30^\circ, which is outside the range
  4. sin is negative between 180^\circ and 360^\circ: 180^\circ+30^\circ=210^\circ and 360^\circ-30^\circ=330^\circ
  5. x=23.6^\circ,\ 156.4^\circ,\ 210^\circ,\ 330^\circ

More on this topic: Solving Trigonometric Equations worksheet with full solutions

All AQA Level 2 Further Maths practice questions

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