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## Get ready for AP® Calculus

### Course: Get ready for AP® Calculus > Unit 7

Lesson 4: Sinusoidal equations# Solving cos(θ)=1 and cos(θ)=-1

Sal solves the equations cos(θ)=1 and cos(θ)=-1 using the graph of y=cos(θ). Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- What is exactly happening at1:20and why is 'y' taken as equal to cosQ?(28 votes)
- im learning this myself and this is what i have so far.

"y" is taken as the equal to cos(theta) because that is what you are measuring. remember the "x" axis is actually the "theta" axis. you you are graphing where cos(theta) is when theta is at a specified value. so x-axis=theta and y-axis=cos(theta). if you were trying to find the sin(theta), this would be a different graph, then that would make y-axis=sin(theta).

i was at first confused from the fact that the coordinates were supposed to be (cos, sin), but then realized that this only applied to the unit circle.

at1:20he is showing the direction you should travel on the unit circle to get the correct coordinates for the graph. if you move the wrong direction, you aren't getting the correct numbers. you can really tell when you start graphing.(48 votes)

- Is there some sort of intuition as to why cos is the x of the unit circle and sin is the y?(15 votes)
- FishHead,

Well, yes, it's all based on how we define the angle that we are interested in when we are using the unit circle (most people call that angle θ, theta). Just as a convention, we define θ as the angle from the X axis, to the line that goes from the origin to the point on the circle we are interested in. So when we talk about the point on the unit circle at 45 degrees, ( or pi/4 radians), that's 45 degrees from the positive X axis.

If we have some point on the unit circle, and we draw a line from the origin to that point, that gives us 1 leg of a triangle. If we use the X axis as the second leg of the triangle, and then draw a line that goes through the point on the circle to the x axis (and we make sure the line is perpendicular to the x axis), that gives us the third leg of our right triangle. If you do this, you'll see that the angle, θ, that I was talking about before is one of the three angles of this right triangle. It is the angle at the point of the triangle that touches the origin. So if we take the cosine of that angle, that is equal to the adjacent side of that triangle divided by the hypotenuse. But notice that the adjacent side of the triangle is just the x coordinate for the original point on the unit circle that we are interested in, and the hypotenuse of the circle is just 1. So the adjacent side over the hypotenuse is just the x coordinate of the point on the circle divided by 1.

Similarly, the sign of the angle θ is the opposite side of the triangle that we drew, divided by the hypotenuse. But this is just the y coordinate of the point on the circle, divided by 1… which is just the y-coordinate of the point on the circle.

If this doesn't make sense, it may help to draw a pair of x and y axes, and draw a circle on the paper centered on the origin with a radius of 1. Then, pick a point on the circle (and I would pick a point in quadrant 1 until you get comfortable with the concepts), and draw the appropriate right triangle.(31 votes)

- I wonder does cosθ² + sinθ² = 1?(4 votes)
- If you mean:

cos (θ²) + sin(θ²), then that is NOT equal to 1,

except for a few special angles such as θ=√(2π), θ=0 or θ= ½√(2π)

If you mean: (cos θ )² + (sin θ)² = 1

Which is usually written as: cos² (θ) + sin²( θ) = 1

Then that is true.(27 votes)

- It's a greek letter which often represents an angle.Like a variable(10 votes)

- sal can u make another video on this? The problems in the exercises are harder than the videos(7 votes)
- I get how to do the problems in the next exercise however Im getting stumped on how cos-1(.35) = 1.21 My calculators say 69.512. When I try to convert this to radians it doesnt add up either.(1 vote)

- Why don't we have any lessons on Secant, Cosecant and Cotangent?

I have no visual idea of what they are.(3 votes)- I think there are some lessons... did you use the search bar to find lessons?

If not, give it a try. The search bar is on all KA screens in upper left.(3 votes)

- Why does the sine and cosine graph range between 1 and -1(2 votes)
- Because sin = Opposite/ Hypotenuse and cos = Adjacent/ Hypotenuse

and hypotenuse of a triangle is always greater than to the other two sides. So the fraction of opp./hyp. or adj./hyp. can never be greater than 1(3 votes)

- At2:05about it states that when cos= 1 then theta is 0; I'm confused, I thought cos equalled one at 2π.(3 votes)
- This is also the same for every even multiple of pi. Ex: 4pi would be twice around the circle, 8pi would be four times around.(1 vote)

- In the exercise after this video, when the question asked to choose all the values for the equation sin (0.65)= x, the answer told to find the values between x is greater than -π and less than or equal to π. Why didn't it say greater than or equal to -π ?(2 votes)
- What you are saying implies that you are looking for answers from -pi to infinity (there would be an infinite number of answers). You need constraints placed upon both ends.(3 votes)

- Is cos 180 = -1 or 1

In my math text book it says 1 but the internet says -1(1 vote)- It would be -1.

Visualize the unit circle. An angle of 180 degrees would place you at (-1,0). Thus, cos 180 = -1.(5 votes)

## Video transcript

- [Instructor] In the graph
below, for what values of theta does cosine of theta equal one, and for what values of theta does cosine of theta equal negative one? And they've very nicely graphed it for us, the horizontal axis is the theta axis, and the vertical axis is the y-axis, and so this is the graph of y
is equal to cosine of theta. And it makes sense with
a unit circle definition, and I'll just make sure that
we're comfortable with that, because with our unit circle definition, so let me draw ourselves a unit circle, and I'm just going to
draw it very roughly, just so that we get the general idea of what's going on here. When theta is equal to
zero, we're at this point right over here on the unit circle. Well what's the
x-coordinate of that point? Well it's one, and you see
when theta's equal to zero, on this graph, cosine of
theta is equal to one. When theta is equal to pi over two, we're at this point on the unit circle, and the x-coordinate is what? Well, the x-coordinate there is zero. And you see once again,
when we're at pi over two, the x-coordinate is zero, so
this is completely consistent with our unit circle definition. As we move in the rightward direction, we're moving counterclockwise
around the unit circle, and as we move in the leftward direction, we're moving counter, sorry. If we move in the rightward direction, we're moving counterclockwise, and as we're moving in
the leftward direction along the axis in the negative angles, we're moving in the
clockwise, we're moving in the clockwise direction
around our unit circle. So let's answer their question. For what values of theta does
cosine of theta equal one? Well we can just read the
graph right over here. It equals one, so cosine
of theta equals one, cosine of theta equals one at, at theta is equal to, well,
we see it right over here. Theta is equal to zero, theta is equal to, well we've gotta go all
the way again to two pi, two pi, but then it just keeps going on and on, and it makes sense. Theta equaled, or sorry, cosine of theta, the x-coordinate on this
unit circle equaled one right when we were at zero angle, and we had to go all the
way around the circle to get back to that point, two pi radians. But then it'll be again when
we get to four pi radians, and then six pi radians,
so two pi, four pi, six pi, and I guess you could
see the pattern here. We're gonna keep hitting
cosine of theta equals one every two pi, so you could
really kind of view this as every multiple of two pi. Two pi n, where n is an
integer, n is integer... is an integer. And that applies also for negative values. If you're going the other way around, if we're going the other way around, we don't get back until
we get to negative two pi. Notice we were at zero,
and then the next time we're at one again is at negative two pi, and then negative four pi, and then over and over and over again. But this applies, if n
is an integer, n can be a negative number, and so we get to all of the negative values of theta where cosine of
theta is equal to one. Now let's think about when cosine of theta is equal to negative one. So cosine of theta is
equal to negative one at theta is equal to,
well we can just look at this graph right over here. Well when theta is equal to pi, when theta is equal to pi, and let's see, well, it kinda goes off this graph, but this graph would keep going like this, would keep going like this, and you'd see it would also be at three pi. And you can visualize it over here. Theta, cosine of theta
is equal to negative one when we're at this point
on the unit circle. So that happens when we get to pi radians, and then it won't happen again until we get to two pi, three
pi radians, three pi radians. And it won't happen again
until we go to two pi, until we add another two pi, until we make one entire revolution, so then that's going to
be five, five pi radians. And you can keep going on and on and on, and that's also true in
the negative direction, so if we take two pi away from this, so if we were here, and if
we go all the way around back to negative pi, it
should also be the case, and you actually see it
right over here on the graph. So you could think about this as two pi, two pi n plus pi, or you could view it as two n plus one, or two n plus one times pi, where pi is, sorry, where n is an integer. Let me right that a little
bit neater, n is integer. At every one of those points, cosine, or for every one of these
thetas, cosine of theta is going to keep hitting negative
one over and over again. And you see it, it goes
it goes from one bottom, where you can kind of
valley to the next valley, it takes two pi to get to the next valley, two pi to get to the next valley. And that was also the
same thing for the peaks. It took two pi to go
from the top of one hill to the top of the next,
and then two pi again to the top of the hill after that.