Solving Trig Equations

Find every solution of a trig equation in a given range, in degrees or radians: compound angles, factorising, and using identities to turn equations into quadratics.

In the exam for Edexcel Further Pure Maths 4PM1, IGCSE Additional Maths 0606 · 43 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

One equation, many answers

Solve $sin x = 0.5$ and your calculator says $3 0^{\circ}$ . But $15 0^{\circ}$ works too, and so does $39 0^{\circ}$ . A trig equation usually has many solutions, because the wave keeps repeating.

Rule

So the question always gives a range, such as $0^{\circ} \leq x \leq 36 0^{\circ}$ . Your job is to find every solution inside it, and none outside it.

The calculator value, $α$ , is the first one. The circle's symmetry gives a second one:

Formula

sin : x = α and x = 18 0^{\circ} - α

$α$ — the calculator value $sin^{- 1} k$ . In radians, use $π - α$ .

Formula

cos : x = α and x = 36 0^{\circ} - α

$α$ — the calculator value $cos^{- 1} k$ . In radians, use $2 π - α$ . You can also use $- α$ .

Formula

tan : x = α and x = α + 18 0^{\circ}

$α$ — the calculator value $tan^{- 1} k$ . In radians, use $α + π$ .

Rule

Then add or take away $36 0^{\circ}$ (or $2 π$ ) to any answer until you have every value in the range. For tan you can step by $18 0^{\circ}$ (or $π$ ).

Worked example

1
Solve $sin x = - 0.6$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
2
$α = sin^{- 1} (- 0.6) = - 36.8 7^{\circ}$ — The calculator value. It is outside the range, but keep it to build the others.
3
$18 0^{\circ} - (- 36.8 7^{\circ}) = 216.8 7^{\circ}$ — The sine partner.
4
$- 36.8 7^{\circ} + 36 0^{\circ} = 323.1 3^{\circ}$ — Add a full turn to bring the first value into the range.
5
$x = 216. 9^{\circ}, 323. 1^{\circ}$ — Both are in the third and fourth quadrants, where sin is negative. Good.

Worked example

1
Solve $cos x = - \frac{1}{2}$ for $0 \leq x \leq 2 π$ , without a calculator.
2
$α = \frac{2 π}{3}$ — $cos \frac{π}{3} = \frac{1}{2}$ , and cos is negative in the second quadrant: $π - \frac{π}{3}$ .
3
$2 π - \frac{2 π}{3} = \frac{4 π}{3}$ — The cosine partner.
4
$x = \frac{2 π}{3}, \frac{4 π}{3}$ — Both are in the range.

Rule

Give angles in degrees to 1 decimal place, and radians to 3 significant figures, unless the question asks for exact answers. On 0606 Paper 1 there is no calculator, so expect exact values.

Common mistake

Stopping at the calculator value. It is often not even in the range. Always look for the partner, then add or take away full turns.

Key idea

The London Eye takes about 30 minutes to turn once. Your seat passes any chosen height twice each turn: once going up, once coming down. That is why $sin x = k$ usually has two answers in every $36 0^{\circ}$ .

Compound angles

Now the angle inside is not just x, as in $cos (2 x + 4 0^{\circ}) = 0.5$ . Treat the whole inside as one angle. The trick is to change the range first.

Steps

1
1. Call the inside angle A, for example $A = 2 x + 4 0^{\circ}$ .
2. Change the range: do to the range exactly what is done to x.
3. Solve for A, finding every value in the new range.
4. Turn each value of A back into x.

Worked example

1
Solve $cos (2 x + 4 0^{\circ}) = 0.5$ for $0^{\circ} \leq x \leq 18 0^{\circ}$ .
2
$4 0^{\circ} \leq 2 x + 4 0^{\circ} \leq 40 0^{\circ}$ — Double the range, then add 40°.
3
$α = cos^{- 1} (0.5) = 6 0^{\circ}$ — The first value of the inside angle.
4
$36 0^{\circ} - 6 0^{\circ} = 30 0^{\circ}$ — The cosine partner. The next ones, 420° and −60°, are outside 40° to 400°.
5
$2 x + 4 0^{\circ} = 6 0^{\circ} \Rightarrow x = 1 0^{\circ}$ — Take away 40°, then halve.
6
$2 x + 4 0^{\circ} = 30 0^{\circ} \Rightarrow x = 13 0^{\circ}$ — Same for the second value.

Worked example

1
Solve $tan 3 x = 1$ for $0^{\circ} \leq x \leq 18 0^{\circ}$ .
2
$0^{\circ} \leq 3 x \leq 54 0^{\circ}$ — Triple the range.
3
$3 x = 4 5^{\circ}, 22 5^{\circ}, 40 5^{\circ}$ — Start at 45° and keep adding 180°. The next, 585°, is too big.
4
$x = 1 5^{\circ}, 7 5^{\circ}, 13 5^{\circ}$ — Divide each by 3.

Worked example

1
Solve $sin (x - \frac{π}{4}) = \frac{1}{2}$ for $0 \leq x \leq 2 π$ . Give exact answers.
2
$- \frac{π}{4} \leq x - \frac{π}{4} \leq \frac{7 π}{4}$ — Take $\frac{π}{4}$ from both ends of the range.
3
$x - \frac{π}{4} = \frac{π}{6}, \frac{5 π}{6}$ — $sin \frac{π}{6} = \frac{1}{2}$ , and the partner is $π - \frac{π}{6}$ .
4
$x = \frac{5 π}{12}, \frac{13 π}{12}$ — Add $\frac{π}{4}$ to each: $\frac{π}{6} + \frac{π}{4} = \frac{5 π}{12}$ and $\frac{5 π}{6} + \frac{π}{4} = \frac{13 π}{12}$ .

Common mistake

Not changing the range. With $tan 3 x = 1$ , keeping $0^{\circ}$ to $18 0^{\circ}$ for 3x finds only $x = 1 5^{\circ}$ and loses two answers.

Rule

The last accuracy mark is lost if you leave an extra answer that is inside the range but wrong. Check every answer against the range for x.

Key idea

Change the range, solve for the inside, then undo the inside. Multiples of x always mean more answers.

Identities and factors

Harder equations mix ratios, such as $cos^{2} x$ with $sin x$ . Use an identity so only one ratio is left. Often you get a quadratic you can factorise.

Rule

Use $sin^{2} x + cos^{2} x = 1$ . 0606 and 4037 print it in the List of formulas. 4PM1 does not: learn it.

Worked example

1
Solve $2 cos^{2} x + 3 sin x - 3 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
2
$2 (1 - sin^{2} x) + 3 sin x - 3 = 0$ — Swap cos² x for 1 − sin² x, so only sin is left.
3
$2 sin^{2} x - 3 sin x + 1 = 0$ — Expand, tidy, and multiply by −1 so the squared term is positive.
4
$(2 sin x - 1) (sin x - 1) = 0$ — Factorise, as if sin x were a letter like s.
5
$sin x = \frac{1}{2} \Rightarrow x = 3 0^{\circ}, 15 0^{\circ}$ — The calculator value and its sine partner.
6
$sin x = 1 \Rightarrow x = 9 0^{\circ}$ — The top of the sine wave: only one answer here.

Some equations factorise straight away, because the same ratio is in every term.

Worked example

1
Solve $2 sin x cos x = sin x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
2
$2 sin x cos x - sin x = 0$ — Bring everything to one side. Do not divide by sin x.
3
$sin x (2 cos x - 1) = 0$ — Take out the common factor sin x.
4
$sin x = 0 \Rightarrow x = 0^{\circ}, 18 0^{\circ}, 36 0^{\circ}$ — Where the sine wave crosses the axis.
5
$cos x = \frac{1}{2} \Rightarrow x = 6 0^{\circ}, 30 0^{\circ}$ — The calculator value and 360° − 60°.

When sin and cos appear once each, with no squares, divide by $cos x$ to get tan.

Worked example

1
Solve $3 sin x = 4 cos x$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
2
$\frac{s i n x}{c o s x} = \frac{4}{3}$ — Divide both sides by 3 cos x. This is safe: sin x and cos x are never 0 at the same angle, so cos x = 0 gives no solution here.
3
$tan x = \frac{4}{3}$ — Sin over cos is tan.
4
$x = 53. 1^{\circ}, 233. 1^{\circ}$ — 53.13° from the calculator, then add 180°.

Common mistake

Dividing both sides by $sin x$ . In $2 sin x cos x = sin x$ that throws away $sin x = 0$ , and with it three answers. Factorise instead.

Common mistake

Giving answers to an impossible equation. If a factor gives $sin x = 1.5$ or $cos x = - 2$ , write that it has no solutions: sin and cos are always between −1 and 1.

Key idea

One ratio only, then solve it like a quadratic. Take out common factors; never divide them away.

sec, cosec and cot

0606 and 4037 also set equations with sec, cosec and cot. 4PM1 students can skip this step. Turn them into cos, sin and tan, then solve as before.

Worked example

1
Solve $2 sec 2 θ = 5$ for $0 \leq θ \leq π$ , where $θ$ is in radians.
2
$sec 2 θ = \frac{5}{2} \Rightarrow cos 2 θ = \frac{2}{5}$ — Sec is 1 over cos, so flip the fraction.
3
$0 \leq 2 θ \leq 2 π$ — Double the range.
4
$2 θ = 1.1593, 2 π - 1.1593 = 5.1239$ — The calculator value in radians, and its cosine partner.
5
$θ = 0.580, 2.56$ — Halve each, then round to 3 significant figures.

When you see $tan^{2}$ with $sec$ , or $cot^{2}$ with $cosec$ , use the matching identity to leave one function.

Worked example

1
Solve $2 tan^{2} x + sec x - 1 = 0$ for $0^{\circ} \leq x \leq 36 0^{\circ}$ .
2
$2 (sec^{2} x - 1) + sec x - 1 = 0$ — From 1 + tan² x = sec² x, so tan² x = sec² x − 1.
3
$2 sec^{2} x + sec x - 3 = 0$ — Expand and collect.
4
$(2 sec x + 3) (sec x - 1) = 0$ — Factorise.
5
$sec x = - \frac{3}{2} \Rightarrow cos x = - \frac{2}{3}$ $\Rightarrow x = 131. 8^{\circ}, 228. 2^{\circ}$ — Flip, then 131.81° and 360° − 131.81°.
6
$sec x = 1 \Rightarrow cos x = 1$ $\Rightarrow x = 0^{\circ}, 36 0^{\circ}$ — Both ends of the range count.

Worked example

1
Solve $3 cot x = tan x$ for $0^{\circ} < x < 18 0^{\circ}$ .
2
$\frac{3}{t a n x} = tan x$ — Cot is 1 over tan.
3
$tan^{2} x = 3$ — Multiply both sides by tan x.
4
$tan x = \pm 3$ — Both square roots count.
5
$x = 6 0^{\circ}, 12 0^{\circ}$ — tan 60° = √3. For −√3: tan is negative in the second quadrant, 180° − 60°.

Rule

Mark schemes give a method mark for using the correct identity, another for solving the quadratic, then accuracy marks for the angles.

Common mistake

Using the value of sec as if it were cos. $sec x = 2$ means $cos x = \frac{1}{2}$ , not $cos x = 2$ .

Common mistake

Forgetting the minus root. $tan^{2} x = 3$ gives $tan x = 3$ and $tan x = - 3$ .

Key idea

Turn sec, cosec and cot upside down into cos, sin and tan. Then it is an equation you already know how to solve.

Which method? Read the question

Look at the equation before you start. Its shape tells you the plan.

You see	You do
$sin x = k$ , $cos x = k$ or $tan x = k$	Calculator value, its partner, then add or take away full turns
$sin (2 x + 3 0^{\circ})$ or $cos \frac{x}{2}$	Change the range first, solve for the inside, then for x
$sin^{2} x$ or $cos^{2} x$ mixed with the other ratio	$sin^{2} x + cos^{2} x = 1$ , then a quadratic
The same ratio in every term	Take it out as a factor; never divide by it
$a sin x = b cos x$	Divide by $cos x$ to get $tan x$
$sec$ , $cosec$ , $cot$ (0606 and 4037)	Flip to cos, sin, tan; use $sec^{2} = 1 + tan^{2}$ or $cosec^{2} = 1 + cot^{2}$
“Hence solve” after a proof	Use the result you just proved to simplify first

Common mistake

Mixing units. If the range is in radians, every answer must be in radians, with the calculator in radian mode.

Key idea

Make it one ratio of one angle. Then: calculator value, partner, full turns, and check the range.

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