Why does cos θ, which normally equals x, become 0 when you evaluate this function

In summary, when evaluating the function for π/r radians, cos θ becomes 0 because the x value on the circle is 0, making the cosine of π/2 and -π/2 equal to 0. This is because for any point (x,y) on the circle, cos θ is equal to x/1, and in this case, the x value is 0.
  • #1
land_of_ice
137
0
Why does cos θ, which normally equals x, become 0 when you evaluate this function for π/r radians?
(π/r means pi over r radians)
Why does cos θ, which normally equals x, become 0 when you evaluate this function for π/r radians?
It says that the point on the circle for this question is (0,1) so zero = x and y = 1 here.
 
Physics news on Phys.org
  • #2
For [tex]x=cos\theta[/tex], when x=0, [tex]\theta=\pi/2[/tex] (it can also be [tex]-\pi/2,3\pi/2[/tex] etc. but I don't think you need to deal with that right now).

When you're looking at the circle and any point on the circle (x,y) remember that [tex]cos\theta[/tex] is adjacent/hypotenuse or x/1=x. So since the point on the circle is (0,1), the x value is 0 so [tex]cos(\pi/2)=0[/tex]
 

Related to Why does cos θ, which normally equals x, become 0 when you evaluate this function

1. Why does cos θ become 0 when θ is equal to 90 degrees?

When evaluating the function cos θ, we are finding the ratio of the adjacent side to the hypotenuse in a right triangle where θ is the angle between the adjacent side and the hypotenuse. When θ is equal to 90 degrees, the adjacent side is equal to 0, meaning the ratio is also equal to 0 and therefore cos θ becomes 0.

2. Why does cos θ become 0 when θ is equal to 270 degrees?

Similar to the first question, when θ is equal to 270 degrees, the adjacent side is now opposite to the hypotenuse in the right triangle. The opposite side has a length of 0, making the ratio of adjacent to hypotenuse also equal to 0 and therefore cos θ becomes 0.

3. Can cos θ ever be equal to 0 for values other than 90 and 270 degrees?

Yes, cos θ can be equal to 0 for values other than 90 and 270 degrees. In fact, there are infinite values of θ where cos θ can be equal to 0. This occurs when the adjacent side and hypotenuse have the same length, creating a right angle in the triangle and resulting in a ratio of 0.

4. Why is cos θ equal to 0 in the first and third quadrants of the unit circle?

In the unit circle, the x-coordinate of a point is equal to cos θ and the y-coordinate is equal to sin θ. In the first quadrant, both coordinates are positive and therefore cos θ is not equal to 0. However, in the third quadrant, the x-coordinate (cos θ) is negative while the y-coordinate (sin θ) is positive. This results in a ratio of 0 for cos θ, making it equal to 0 in the third quadrant.

5. How does the value of θ affect the value of cos θ?

The value of θ directly affects the value of cos θ. As θ increases from 0 to 90 degrees, cos θ also increases from 1 to 0. As θ continues to increase from 90 to 180 degrees, cos θ decreases from 0 to -1. This pattern continues as θ increases, resulting in a periodic function with a range of -1 to 1.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
21
Views
1K
  • Precalculus Mathematics Homework Help
Replies
9
Views
2K
  • Precalculus Mathematics Homework Help
Replies
4
Views
3K
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
  • Calculus
Replies
29
Views
996
  • Precalculus Mathematics Homework Help
Replies
20
Views
3K
  • Precalculus Mathematics Homework Help
Replies
8
Views
3K
Replies
4
Views
623
  • Introductory Physics Homework Help
Replies
26
Views
2K
Replies
2
Views
309
Back
Top