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## Class 11 math (India)

### Course: Class 11 math (India)>Unit 3

Lesson 6: Trigonometric 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 at and why is 'y' taken as equal to cosQ? •  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.

at he 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.
• Is there some sort of intuition as to why cos is the x of the unit circle and sin is the y? •  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.
• I wonder does cosθ² + sinθ² = 1? •  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.
• What is Theta? • sal can u make another video on this? The problems in the exercises are harder than the videos • I dont understand when to use

cos(theta + 360) or cos(theta +180) • 1.The equation cos(theta) = cos(theta + 360°) means that no matter how many complete rotations of 360° you add to the angle theta, it will still have the same cosine value. So, all the possible theta values that give you the same cosine result can be written as theta + 360° multiplied by a whole number (n).

2.The equation cos(theta + 180°) = negative cos(theta) means that if you add 180° to an angle theta, the cosine of the new angle will be the negative of the cosine of the original angle. However, if you add 180° again, this relationship doesn't hold. To maintain the relationship, you need to add 360° instead. So, you get theta + 180° + 360° multiplied by a whole number (n).
• Why don't we have any lessons on Secant, Cosecant and Cotangent?
I have no visual idea of what they are. • Why does the sine and cosine graph range between 1 and -1   