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 value. (Try it on your calculator - see what it says if you try to find the value of tan 90°.)

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

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, called the reference angle, defined as:

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

Quadrant II

Example:

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

math expression

Example:

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

math expression

Example:

θ = 315° = 360° - 45°

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

In each case,

θ is the angle of interest

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

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

Example

Find 2 angles whose cosine is 0.7.

Solution

As we know, there are an infinite number of answers. We will find the first 2 positive angles whose cosine is 0.7.

From our calculator, we have one answer: cos^-10.7 = 45.57°.

Since this is an acute angle, we can use it as the reference angle and write α = 45.57°.

To find another answer, we remember that the 2 places where cosine is positive are the first (I) and fourth (IV) quadrants. The corresponding angle in the fourth quadrant will be:

360° − α = 360° − 45.57° = 314.43°

Check: cos 45.57° = 0.7000373 and

cos 314.43° = 0.7000373

Of course, there are an infinite number of angles where the cosine of the angle is 0.7. Here are some of them (the values where the red arrows are pointing):

math expression

Try the values in your calculator (eg cos 1125.6°) to convince yourself that they are correct. (Your answer rounded to one decimal place should be 0.7 for each one.)

Checking your answers helps you to understand what is going on and it is great for building confidence in mathematics!

Exercises:

1. Write using a positive acute angle:

a) cos190°

b) cot 290°

Answer: a) cos190° = cos(180° + 10°) = -cos(10°)

[Quadrant III, so negative]

b) cot 290° = cot(360° − 70°) = -cot(70°)

[Quadrant IV, so negative]

2. tan 152.4° = [use calculator]

Answer: tan 152.4° = -0.5223

3. csc 194.82° = [use calculator]

Answer: csc 194.82° = 1/(sin 194.82°) = -3.9096

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

Answer: Let sin α = 0.8480, then we have

α = arcsin 0.8480 = 58.0°

Since sin θ is negative, θ must be in Quadrants III or IV.

So θ = 180° + 58.0° = 238.0° or

θ = 360° − 58.0° = 302.0°

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

Students always get confused with sin^-1x and csc x (they are NOT the same: sin^-1x means "the angle whose sine is x", whereas csc x means "1/sin x").
It is standard in science and engineering to use "arcsin".
Most computer programs use "arcsin" or "asin" for the inverse of sine - not "sin^-1", so it is good practice to get used to it.

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

Answer: The only quadrant where tan θ is negative and csc θ is positive is the second quadrant.

Let tan α = 0.809 then α = arctan 0.809 = 39.0°

So θ = 180° − 39.0° = 141.0°

So cos θ = -0.777

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

Answer: The only quadrant where sec θ is positive and sin θ is negative is Quadrant IV.

Let sec α = 1.122, then cos α = 1/1.122 = 0.8912656

So α = arccos 0.8912656 = 27.0°

So θ = 360° − 27.0° = 333.0°

So cot θ = 1/(tan333.0°) = -1.965

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