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Trig Equations0%

Trigonometry · Topic 6 of 7

Trig Equations

Video lesson5 worked examples

Theory

You must be able to solve trigonometric equations for a given domain (usually $0^\circ \le x \le 360^\circ$ ).

There are usually two solutions. The CAST diagram (or the symmetry of trigonometric graphs) is used to find the related angles in different quadrants depending on whether the trig ratio is positive or negative.

Related Values

You may be given one value — say $\sin 40^\circ = 0.643$ — and asked to write down a related value without a calculator, or to put several values in order of size. Both use the same symmetry:

\sin(180^\circ - x) = \sin x \qquad \sin(180^\circ + x) = -\sin x \qquad \sin(360^\circ - x) = -\sin x

\cos(360^\circ - x) = \cos x \qquad \cos(180^\circ - x) = -\cos x

In practice you do not need to memorise these as formulas — read them off the CAST diagram or the shape of the graph. CAST tells you the sign in each quadrant, and the related acute angle tells you the size.

The Golden Rule: first rearrange to get $\sin x$ , $\cos x$ or $\tan x$ on its own. Take the inverse of the positive value to get the base angle, then use the sign to decide which two quadrants (via CAST) the solutions fall in.

⚠️ Common Examiner Traps

Base angle from the positive value: always take $\sin^{-1}$ etc. of the positive number, then place the answers using the sign.
Which quadrants: positive sine → Q1 & Q2; negative cosine → Q2 & Q3; positive tan → Q1 & Q3, and so on. CAST keeps this straight.
Both solutions: the domain $0^\circ$ to $360^\circ$ almost always gives two answers — don't stop at one.
Rearrange fully first: deal with the number in front and the constant before touching the inverse function.

Worked examples

Example 1

Solve $2\sin x - 1 = 0$ for $0^\circ \le x \le 360^\circ$ .

Step 1: Rearrange to isolate $\sin x$ :

2\sin x = 1 \Rightarrow \sin x = 0.5

Step 2: Find the base angle (Quadrant 1):

x = \sin^{-1}(0.5) = 30^\circ

Step 3: Since sine is positive, the second solution is in Quadrant 2 (S): $180^\circ - 30^\circ$ .

Answer: $x = 30^\circ$ and $x = 150^\circ$ .

Example 2

Solve $5\cos x + 2 = 0$ for $0^\circ \le x \le 360^\circ$ .

Step 1: Rearrange:

5\cos x = -2 \Rightarrow \cos x = -0.4

Step 2: Find the related base angle (ignoring the negative sign):

\cos^{-1}(0.4) \approx 66.4^\circ

Step 3: Since cosine is negative, the solutions lie in Quadrants 2 (S) and 3 (T): $180^\circ - 66.4^\circ$ and $180^\circ + 66.4^\circ$ .

Answer: $x = 113.6^\circ$ and $x = 246.4^\circ$ .

Example 3

Solve $\tan x - 3 = 0$ for $0^\circ \le x \le 360^\circ$ .

Step 1: Rearrange:

\tan x = 3

Step 2: Find the base angle (Quadrant 1):

x = \tan^{-1}(3) \approx 71.6^\circ

Step 3: Since tangent is positive, the second solution is in Quadrant 3 (T): $180^\circ + 71.6^\circ$ .

Answer: $x = 71.6^\circ$ and $x = 251.6^\circ$ .

Example 4

Related Values

Given that $\sin 35^\circ = 0.574$ , write down the value of (a) $\sin 145^\circ$ and (b) $\sin 215^\circ$ .

Step 1 (a): $145^\circ$ is in the second quadrant, where sine is positive (the S in CAST). Its related acute angle is $180^\circ - 145^\circ = 35^\circ$ .

So the size is the same and the sign is positive:

\sin 145^\circ = 0.574

Step 2 (b): $215^\circ$ is in the third quadrant, where only tangent is positive — so sine is negative. Its related acute angle is $215^\circ - 180^\circ = 35^\circ$ .

\sin 215^\circ = -0.574

Example 5

🎯 Exam-style (order of size)

Write the following in order of size, starting with the smallest. Justify your answer.

\cos 100^\circ, \qquad \cos 20^\circ, \qquad \cos 190^\circ

Step 1: Use CAST to get the sign of each. Cosine is positive only in the first and fourth quadrants:

$\cos 20^\circ$ — first quadrant, so positive.
$\cos 100^\circ$ — second quadrant, so negative.
$\cos 190^\circ$ — third quadrant, so negative.

Step 2: Separate the two negatives. Their related acute angles are $180^\circ - 100^\circ = 80^\circ$ and $190^\circ - 180^\circ = 10^\circ$ . Since $\cos 10^\circ$ is larger than $\cos 80^\circ$ , the negative of it is smaller — so $\cos 190^\circ$ is the smallest.

Answer: $\cos 190^\circ,\ \cos 100^\circ,\ \cos 20^\circ$ — the two in the second and third quadrants are negative and the first-quadrant one is positive, so it is largest.

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14 Trig Equations questions with answers and video solutions