Defining and Evaluating the Basic Trig Functions
Objective
ANGLES OF ROTATIONS
So, before we really start anything in the trigonometry unit, we need to talk about how angles work.
To start, let’s say we have a graph:
y
x
Now, the way we measure an angle, is we start at the first quadrant
And we rotate around the graph until we get to where we want.
For example:
As we can see, we started in the first quadrant
And we rotated to the second quadrant.
Now, some things to remember:
A full rotation around the graph is a 360 degree rotation
And a rotation around each quadrant is 90 degrees.
So, since we rotated around to two quadrants
This angle would be 180 degrees.
SO…..THAT’S IT?
Well, kind of.
That’s the easiest way we can look at it
But, it does get a little more complicated.
So far, the kind of example we’ve gone over is a positive one.
Because we rotated to the left.
Now, this can get pretty confusing at times to see
So, what we do is use the right hand rule to measure positive or negative angles.
So what’s the right hand rule?
Well, you take your right hand
And curl your fingers while sticking your thumb out.
If your thumb is pointing towards you
It’s a positive rotation
If your thumb is pointing away from you
It’s a negative rotation
So why is this important?
Well, because of the different ways you can create angles.
COTERMINAL ANGLES:
So, now that we understand the right hand rule, and the difference between positive rotation, and negative rotation
Let’s talk about coterminal angles.
So, a coterminal angle is:
“An angle that shares the same terminal side.”
So what’s a terminal side?
Well it’s easier to show than explain
So let’s look at another graph:
Now, where we started our angle is called the initial side
Initial side
terminal side
The side of the angle where it stopped rotating is called the terminal side.
So why does this matter?
Well, because we can have angles like:
We know this is a positive rotation
But what about a negative rotation?
Like:
In this case, the coterminal angle is -103 degrees.
But that’s not all!
What about:
What if instead of going negative
We rotate completely around again?
Like:
They still share the same terminal side
So these are coterminal angles as well.
RADIANS
So what is a radian?
Well, a radian is another way to measure a degree, but in a way that makes better sense when dealing with circles.
Mainly because since we discovered the circumference of a circle deals with pi, it makes better sense to use pi to measure different parts of a circle.
So how do we get radians?
Well, it’s based off of what we call the unit circle:
-1
1
1
-1
So, the unit circle has a radius of 1 unit
Which means if we try to find the circumference of this circle we can see:
Now, plugging in 1 for r, we get:
So, we can conclude that:
Now let’s go over right triangles:
So, now that we know the equation
We can convert any degree to any radian we need.
The only thing we need to do is set up the proportion.
So for example:
Convert the following angles to radians:
So to convert this, we need to set up the proportions and solve:
This also lets us see different angle measurements
As radians
Like so:
And that’s how we solve for radians.
So, now that we have the equation, let’s look at some examples:
Example 1:
Convert the following radian to degrees
Then, find a positive coterminal angle
And a negative coterminal angle:
So, first we know a full rotation is 2pi
And it’s also 360 degrees
So, we can set this up like:
So our answer is 270 degrees!
Well, let’s look at a 270 degree rotation:
Well, as we can see, a negative rotation of -90 degrees would be a coterminal angle.
So our negative coterminal angle is:
Now a positive coterminal angle won’t happen unless we fully rotate
And since a full rotation is 360 degrees
Then:
Example 2:
Convert the following degrees to radians
Then, find a positive coterminal angle
And a negative coterminal angle:
So, first we know a full rotation is 2pi
And it’s also 360 degrees
So, we can set this up like:
So our answer is 19pi/18!
But how do we see that rotation?
Well, we could try to graph it
But is there a better way?
Well we know that a positive coterminal angle will be a full rotation
Plus our original angle.
Since we know a full rotation is 2pi
Then:
Now, a negative rotation may be tricky
But isn’t it just the opposite of positive?
So since we added with positive
What if we subtract with negative?
Well then, we’d get:
A FEW THINGS TO GO OVER FIRST
So, before we go over the actual lesson, there are a few things to go over first.
Right Triangles – a right triangle is a triangle with a right angle. It’s important because a triangle can only have one right angle (since the combined angles add up to 180 degrees). All other angles are considered acute angles (or less than 90 degrees.)
Congruent Sides – Congruent sides have the same length, and are marked like so:
Congruent Angles – Congruent angles have the same angle measure, and are marked like so:
REVIEW OF THE PYTHAGOREAN THEOREM
This is the Pythagorean theorem:
Where each side of a triangle is: a, b, and c; and where a and b are the sides of the triangle with c as the hypotenuse, then:
Now, we can actually see this is true when looking at an actual triangle with measurements.
Like:
Now let’s plug in these numbers and see what we get:
a = 3
b = 4
c = 5
SO WHAT DOES THIS HAVE TO DO WITH OUR TRIANGLES?
Actually, by using the Pythagorean theorem, we can develop all sorts of things about these triangles.
To start off, let’s look at a 45-45-90 triangle:
Now, let’s explore this triangle a little (no matter how weird that may seem).
Now we know that the opposing sides of the 45 degree angles are congruent,
So, let’s mark that:
Now, again, since we know that those two sides are the exact same length
Then we can actually label them as well, so let’s let the length of those two
Sides be x.
x
x
Now all we are missing is the hypotenuse, so let’s use the Pythagorean Theorem to find it!
Remember, the Pythagorean theorem is:
Since we are looking for the hypotenuse, that means that a = x, and b = x, so what we have is:
And since we are trying to solve for c, let’s take the root!
AND THAT’S ESSENTIALLY IT!
So, now that we can see what the triangle equals, we can basically figure out anything we need when we are given a 45-45-90 degree triangle.
The 30-60-90 Triangle
Now that we’ve reviewed over the easiest special triangle, let’s get into the harder one.
So the 30-60-90 triangle follows the same kind of pattern that the 45-45-90 triangle follows, except it takes a little more work to show.
So, let’s get into it:
Proving the 30-60-90 triangle
So, before you lose it because you saw the “P” word, don’t worry, this PowerPoint will prove it for you.
But it may take a few slides. �So, let’s get started:
First, let’s start off with an equilateral (equal sided) triangle:
Now, we can see that we have all 60 degree angles.
But, to get a 30 degree angle, we need to divide one of those 60 degree angles in half.
So, let’s drop an altitude, which will act like an angle bisector, and we’ll have two 30 degree angles instead.
By dropping the altitude, we created two right angles, AEC, and BEC.
The altitude also bisects the segment, making E the midpoint of AB and dividing AB into two equal segments, AE, and EB, both of which are half of AB.
Or in other words, AE and EB = 1.5 since AE = 3.
Now, what we wanted though, is to come up with a more generic rule, so to help us find out some rules, let’s let:
AE = t
But if, AE = t, and AC is twice as big as AE, then we know that AC = 2t
t
2t
So, now the issue is finding the remaining side to figure out how these triangles work.
Using the Pythagorean Theorem (again)
So, let’s look at what we had before:
t
2t
Using the Pythagorean theorem, we can see then that:
A = t
B = CE (segment CE since we don’t actually know what it is)
C = 2t
So:
And that’s essentially it!
So, now that we can see what the triangle equals, we can basically figure out anything we need when we are given a 30-60-90 degree triangle.
So to start, let’s begin with �SOHCAHTOA
SOHCAHTOA
A really weird acronym that will help you remember the basic trig functions
Let’s break them down:
S – Sine
O- Opposite
H- Hypotenuse
C- Cosine
A- Adjacent
H – Hypotenuse
T- Tangent
O- Opposite
A- Adjacent
So, what does this all mean?
It means this:
What does that mean?
Basically what we mean is when looking at a triangle, the angle that is being measured has certain sides associated to it.
But, this is easier to show than explain, so here is a triangle:
And here is the angle associated with that triangle
Now, the side that is opposite of this angle is:
The side that is adjacent of this angle is:
And of course, the hypotenuse of this triangle is:
So, in SOHCAHTOA, the sine of an angle is the opposite over the hypotenuse, or:
Now let’s go back to �SOHCAHTOA
SOHCAHTOA
A really weird acronym that will help you remember the basic trig functions
Let’s break them down:
S – Sine
O- Opposite
H- Hypotenuse
C- Cosine
A- Adjacent
H – Hypotenuse
T- Tangent
O- Opposite
A- Adjacent
So, what does this all mean?
It means this:
So, to review, which side is what?
Basically what we mean is when looking at a triangle, the angle that is being measured has certain sides associated to it.
But, this is easier to show than explain, so here is a triangle:
And here is the angle associated with that triangle
Now, the side that is opposite of this angle is:
The side that is adjacent of this angle is:
And of course, the hypotenuse of this triangle is:
So, in SOHCAHTOA, the cosine of an angle is the adjacent over the hypotenuse, or:
NOW, FOR THE LAST TIME, LET’S GO BACK TO �SOHCAHTOA
SOHCAHTOA
A really weird acronym that will help you remember the basic trig functions
Let’s break them down:
S – Sine
O- Opposite
H- Hypotenuse
C- Cosine
A- Adjacent
H – Hypotenuse
T- Tangent
O- Opposite
A- Adjacent
So, what does this all mean?
It means this:
ONE LAST LOOK AT THE SIDES.
Basically what we mean is when looking at a triangle, the angle that is being measured has certain sides associated to it.
But, this is easier to show than explain, so here is a triangle:
And here is the angle associated with that triangle
Now, the side that is opposite of this angle is:
The side that is adjacent of this angle is:
And of course, the hypotenuse of this triangle is:
So, in SOHCAHTOA, the tangent of an angle is the opposite over the adjacent, or:
Example 1:
15
20
35
Well, we remember from SOHCAHTOA that:
We can see that the side that is opposite of the angle is 15
And, we can see that the side that is adjacent to the angle is 20
So:
Or:
Example 2:
15
22
12
Well, we remember from SOHCAHTOA that:
We can see that the side opposite the angle is 15
And, we can see that the hypotenuse is the biggest side, which we know is 22
So:
Example 3:
16
27
10
Well, we remember from SOHCAHTOA that:
We can see that the side that is adjacent of the angle is 16
And, we can see that the hypotenuse is the biggest side, which we know is 27
So:
Taking the Sine of Special triangles
So, now that we’ve reviewed over the big picture
And we’ve gone over the trig functions
What happens when we have a special triangle, and we want to find the sine of it?
Like so:
x
x
Now, if we wanted to take the sine of let’s say
45 degrees.
We would need have:
However, we can’t have a radical in the denominator
So:
Now, let’s say we want to find the sine of 30 degrees.
Then let’s use the other special triangle:
t
2t
But what about 60 degrees?
Same thing:
Now, let’s talk about cosine:
Taking the Cosine of Special triangles
So, now let’s talk about cosine
So, just like with sine, what happens when we have a special triangle, and we want to find the cosine of it?
Like so:
x
x
Now, if we wanted to take the cosine of let’s say
45 degrees.
We would need have:
However, we can’t have a radical in the denominator
So:
Now, let’s say we want to find the cosine of 30 degrees.
Then let’s use the other special triangle:
t
2t
But what about 60 degrees?
Same thing:
Now, let’s talk about tangent:
Taking the Tangent of Special triangles
So, now let’s talk about tangent
So, just like with cosine, what happens when we have a special triangle, and we want to find the tangent of it?
Like so:
x
x
Now, if we wanted to take the tangent of let’s say
45 degrees.
We would need have:
Now, let’s say we want to find the tangent of 30 degrees.
Then let’s use the other special triangle:
t
2t
But what about 60 degrees?
Same thing:
However, we can’t have a radical in the denominator
So:
So what does this mean?
THE BIG PICTURE
The big picture is this:
Angle | Sine | Cosine | Tangent |
| | | |
| | | |
| | | |
1
So, what does this have to do with coterminal angles?
Well, this is true for any angle that is coterminal with 45, 30, or 60.
Now before, we were talking about angles that were coterminal with a full rotation
But, when it comes to triangles, we only talk about up to 90 degree angles.
Here’s why:
THE COTERMINAL ANGLES OF TRIG
So, when it comes to trig functions, we deal in 90 degrees
The reason for this is because of the different quadrants. �So let’s review over that for a quick second:
Now, as we can see, we have four quadrants:
I
II
III
IV
And, each quadrant has a specific type of x and y coordinate
For example:
+x, +y
-x, +y
-x, -y
+x, -y
Now, this is important because it shows how angle measures change for sine and cosine
For example, let’s say we have a 30-60-90 triangle in the first quadrant
Then we know that all of the coordinates are positive
But, when we go to the second quadrant
We can see that the side that lays on the x-axis is negative.
So, when we go to the third quadrant:
And finally in the last quadrant:
So, each angle that is coterminal to either 30 or 60 will cause the sine or cosine to be either positive or negative
Depending on which quadrant it is in.
Which leads us to:
QUADRANT ANGLES
Now, since sine, cosine, or tangent will be either positive or negative
Due to what quadrant the angle is in
This lead to some rules about when they are positive
And when they are negative
To start, let’s look at a graph:
Now, let’s place our quadrants:
I
II
III
IV
Now, since x and y are both positive in quadrant 1
Then, we know all of the functions will be positive:
(x, y)
All
But, since x is negative, and y is positive in quadrant 2
The only function that is positive is sine:
(-x, y)
Sine
Now, since both x and y are negative in quadrant 3
The only function that is positive is tangent:
(-x, -y)
Tangent
(x, -y)
Cosine
Finally, since x is positive, but y is negative in quadrant 4
The only function that is positive is Cosine:
There are a few ways to remember this:
You can use the acronym:
All Students Take Classes
Or, you can think of cosine as relating more towards x then y
And Sine relates more towards y than x.
So when y is negative, Sine is negative
When x is negative, Cosine is negative
So how does this help us?
Well, if we know which quadrant the angle is in
And we know what the coterminal angle is
Then we can easily find the sine, cosine, or tangent of the angle.
Example 1:
Evaluate the trigonometric function without using a calculator.
Angles are given in degrees.
So, looking at this angle, we know that it’s not in the first quadrant
It’s in the second quadrant.
Which means for sine, it’s going to stay positive
So, now we need to find a coterminal angle.
However, we know that if you go from the first quadrant
To the second quadrant
You rotate 180 degrees
So, let’s subtract 135 from 180:
So, we’re looking for the sine of 45 degrees
But we know that!
It’s:
Example 2:
Evaluate the trigonometric function without using a calculator.
Angles are given in degrees.
So, looking at this angle, we know that it’s not in the first quadrant
It’s in the second quadrant.
Which means for cosine, it’s going to be negative
So, now we need to find a coterminal angle.
However, we know that if you go from the first quadrant
To the second quadrant
You rotate 180 degrees
So, let’s subtract 120 from 180:
Now, we can see what we’re looking for
We’re looking for the cosine of 60 degrees
So:
But, since the angle is in quadrant 2
And we know that the only trig function that is positive in quadrant 2 is sine
Then our real answer must be:
Example 3:
Evaluate the trigonometric function without using a calculator.
Angles are given in degrees.
So, looking at this angle, we know that it’s not in the first quadrant
It’s in the third quadrant.
Which means for tangent, it’s going to stay positive
So, now we need to find a coterminal angle.
However, we know that if you go from the first quadrant
To the second quadrant
You rotate 270 degrees
So, let’s subtract 225 from 270:
So, we’re looking for the tan of 45 degrees
But we know that!
It’s:
Example 4:
Evaluate the trigonometric function without using a calculator.
Angles are given in degrees.
So, looking at this angle, we know that it’s not in the first quadrant
It’s in the fourth quadrant.
Which means for sine, it’s going to be negative
So, now we need to find a coterminal angle.
However, we know that if you go from the first quadrant
To the fourth quadrant
You rotate 2pi degrees
So, let’s subtract 5/3 pi from 2 pi:
So, we’re looking for the sine of pi/3
But we know that!
It’s:
However we know this is in the fourth quadrant
So, sine needs to be negative
So: