Find the point on the terminal side of θ = three pi divided by four that has a y coordinate of 1. Show your work for full credit.
Choose the point on the terminal side of 135°.
tan(3π/4) = 1/x
x = 1/tan(3π/4) = -1
point is (-1,1)
To find the point on the terminal side of θ = three pi divided by four that has a y-coordinate of 1, we can use the unit circle.
The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the coordinate plane. It is often used to represent angles in trigonometry.
Since the y-coordinate is 1, we know that the point will lie on the unit circle at a height of 1.
To determine the x-coordinate, we can use the cosine function because cosine is equal to the adjacent side divided by the hypotenuse in a right triangle. In this case, the adjacent side will be the x-coordinate and the hypotenuse will be 1 (since it's the radius of the unit circle).
The cosine of an angle can be found by evaluating the expression cos(θ).
Let's calculate it step by step:
1. We are given θ = three pi divided by four. Let's substitute this value into the cosine function: cos(θ) = cos(three pi divided by four).
2. To find the value of cos(three pi divided by four), we can use the reference angle. The reference angle is the angle formed between the terminal side and the x-axis. For θ = three pi divided by four, the reference angle is pi divided by four.
3. The cosine of pi divided by four is equal to the square root of 2 divided by 2. So, cos(pi divided by four) = square root of 2 divided by 2.
4. Now, we can find cos(three pi divided by four) using the property of the cosine function: cos(three pi divided by four) = -cos(pi divided by four) = -square root of 2 divided by 2.
Therefore, the point on the terminal side of θ = three pi divided by four with a y-coordinate of 1 is (-(square root of 2 divided by 2), 1).
Please note that the x-coordinate is negative because the angle θ = three pi divided by four lies in the third quadrant of the coordinate plane, where x-values are negative.
To find the point on the terminal side of θ = (3π/4) with a y-coordinate of 1, we can use the unit circle.
1. Start by drawing the unit circle, which is a circle with a radius of 1 centered at the origin (0, 0).
2. Identify the angle θ = (3π/4). In the unit circle, this angle corresponds to a point on the terminal side.
3. Since we want to find a point with a y-coordinate of 1, look for the point on the unit circle where the y-coordinate is 1.
4. On the unit circle, the y-coordinate is positive above the x-axis and negative below the x-axis. Since we're looking for a y-coordinate of 1, we can focus on the top half of the unit circle.
5. In the top half of the unit circle, find the angle that corresponds to a y-coordinate of 1. This angle turns out to be π/4.
6. Notice that θ = (3π/4) is bigger than π/4, but both angles have the same y-coordinate of 1.
7. Therefore, the point on the terminal side of θ = (3π/4) with a y-coordinate of 1 is the same as the point on the terminal side of θ = π/4 with a y-coordinate of 1.
8. The coordinates of the point on the unit circle with θ = π/4 and y-coordinate 1 are (√2/2, 1).
9. Therefore, the point on the terminal side of θ = (3π/4) with a y-coordinate of 1 is (√2/2, 1).
Note: This point is in the second quadrant of the coordinate plane.