Inspection

First, we need to find an initial solution.

So, for example, let’s say we want to find the values of x when \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}}.

We want to draw the graph, and mark on a horizontal line where the condition is met, i.e. where \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}}.

By inspection, we know that the initial solution is at 30°, and we can see it repeats at every 180°.

We’ll denote this 30° ± 180°n.

CAST Diagrams

The CAST diagram is a handy tool to show us where values of the standard trig functions are positive.

So, here’s the breakdown:

• For 0° < x < 90°, ALL of \textcolor{blue}{\sin x}, \textcolor{limegreen}{\cos x} and \textcolor{red}{\tan x} are positive
• For 90° < x < 180°, ONLY \textcolor{blue}{\sin x} is positive
• For 180° < x < 270°, ONLY \textcolor{red}{\tan x} is positive
• For 270° < x < 360°, ONLY \textcolor{limegreen}{\cos x} is positive

Think back to plotting the Unit Circle, in the Trig Basics section.

Let’s say we want to find the values of x such that \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}}, as before.

We know that there is a solution when x = 30°.

First, we plot the point on the diagram, then find the corresponding angles:

From here, we can see that \textcolor{red}{\tan x} = \textcolor{purple}{\dfrac{1}{\sqrt{3}}} when x = 30° ± 360°n or 210° ± 360°n, or, more concisely, 30° ± 180°n.

We ignored the two solutions where \textcolor{red}{\tan x} is not positive, i.e. 90° < x < 180° and 270° < x < 360°.

Dealing With Trig Transformations

Transformations pose a little bit of a problem… See, CAST diagrams become much harder to navigate now. You’re much better off sketching out the function and solving using the horizontal line technique.

So, let’s just begin with an example.

We have f(x) = \cos 3x. Find the values of x \in \lbrack 0°, 360°\rbrack * such that f(x) = \textcolor{purple}{\dfrac{1}{2}}.

* This is just set notation, meaning 0° \leq x \leq 360°

Well, we have a series of solutions, but they’re not immediately obvious.

What we can do instead is plot the regular \textcolor{limegreen}{\cos x} graph on an interval three times as large as the proposed interval, and divide our solutions there by 3.

\textcolor{limegreen}{\cos x} = \textcolor{purple}{\dfrac{1}{2}} has solutions 60°, 300°, 420°, 660°, 780°, 1020°.

Therefore, \cos 3x = \textcolor{purple}{\dfrac{1}{2}} has solutions 20°, 100°, 140°, 220°, 260°, 340°.