6. Trigonometric Functions of Any Angle

Think About This...

Earlier, in the section Values of Trigonometric Functions, we were given the value of a trigonometric ratio and we needed to find the angle.

The first example we did was:

Find θ, given that tan θ = 0.3462

Using our calculators, we found that θ = tan-10.3462 = 19.096°.

Now we know this is correct because tan 19.096° = 0.3462.

But also tan 199.096° = 0.3462.

Likewise, tan 379.096° = 0.3462. And some negative angles also work: tan (-160.904°) = 0.3462.

What's going on here? They all equal `0.3462`. How many answers are there?

Periodic Functions

Trigonometric functions are periodic, that is, their values re-occur over and over again. You will get a better idea of the periodic nature of trigonometric functions if you check out the chapter on Graphs of The Trigonometric Functions.

What it means is that there are many, many values of θ that will work in the equation tan θ = 0.3462. The number is infinite, actually.

Let's see what the graph of y = tan θ looks like. We see that there is a regular pattern that recurs every `180°`. There are also "gaps" in the curve (at `90°`, `270°`, ...) where tan θ has no finite value. (Try it on your calculator - see what it says if you try to find the value of `tan\ 90°`.)

Graph of tan curve

Now, for our problem above, we can see that there are going to be an infinite number of solutions.

Graph of tangent function - solve

I have drawn a line so we can see where the y-value is `0.3462`. I have then dropped the graph intersection value down on to the θ axis. We see that tan 19° = 0.3462 and we can also see the other values that I suggested before:

tan 199° = 0.3462 and tan 379° = 0.3462.

For the negative angles,

tan (-160°) = 0.3462 and tan (-340°) = 0.3462

How Do we Find All these Angles?

Our problem here is to find a method so that we can find all the values of θ that will work in a given trigonometric equation. We start with the idea of the reference angle.

The Reference Angle

For any angle θ (greater than `90^@`), there is a corresponding acute angle α (alpha), called the reference angle, defined as:

Quadrant II: Written in the form θ = 180° - α

Quadrant II

Angle in second quadrant

Example 1

θ = 130° = 180° - 50°

In this case, 130° is in the second quadrant and has reference angle α = 50°.

Quadrant III: Written in the form θ = 180° + α

Quadrant III

Angle in the 3rd quadrant

Example 2

θ = 240° = 180° + 60°

In this case, 240° is in the 3rd quadrant and the reference angle is α = 60°.

Quadrant IV: Written in the form θ = 360° - α

Quadrant IV

Angle in the 4th quadrant

Example 3

θ = 315° = 360° - 45°

In this case, 315° is in the 4th quadrant and the reference angle is α = 45°.


1. In each case,

θ is the angle of interest

α is an acute angle (that is, less than 90°)

2. We use the idea of reference angle in the following type of question.

Example 4

Find 2 angles whose cosine is `0.7`.


1. Write using a positive acute angle:

a) `cos\190^@`

b) `cot\ 290^@`

2. `tan\ 152.4^@` = [use calculator]

3. `csc\ 194.82^@` = [use calculator]

4. Find θ if sin θ = -0.8480 (0° < θ < 360°)

Arcsin or sin-1?

Why do I use "`arcsin`" instead of what is on your calculator, "`sin^-1`"?

5. If `tan\ θ = -0.809` and `csc\ θ > 0`, find `cos\ θ`.

6. If `sec\ θ = 1.122` and `sin\ θ < 0`, find `cot\ θ`.

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