Question

1. Use the cosine and sine functions to express the exact coordinates of P in terms of angle θ.

2. Mark a point Q on the unit circle where tangent has a value of 0.

Answer 1

To express the coordinates of point P in terms of angle θ, we can use the cosine and sine functions. The exact coordinates of point P in terms of angle θ are (Ax, Ay). To find a point Q on the unit circle where the tangent has a value of 0, we can set the tangent function equal to 0 and solve for the angle. A possible point Q is (1, 0).

Step-by-step explanation:

To express the coordinates of point P in terms of angle θ, we can use the cosine and sine functions. Let the length of the adjacent side be Ax, the length of the opposite side be Ay, and the length of the hypotenuse be A. The cosine of the angle θ is given by cos θ = Ax/A, and the sine of the angle θ is given by sin θ = Ay/A. So the exact coordinates of point P in terms of angle θ are (Ax, Ay).

To find a point Q on the unit circle where the tangent has a value of 0, we can set the tangent function equal to 0 and solve for the angle. The tangent function is defined as tan θ = Ay/Ax. Setting it equal to 0, we get Ay = 0. So the y-coordinate of point Q is 0. For the x-coordinate, we can choose any value on the unit circle. So a possible point Q is (1, 0).

Answer 2

For a point defined bt a radius R, and an angle θ measured from the positive x-axis (like the one in the image)

The transformation to rectangular coordinates is written as:

x = R*cos(θ)

y = R*sin(θ)

Here we are in the unit circle, so we have a radius equal to 1, so R = 1.

Then the exact coordinates of the point are:

(cos(θ), sin(θ))

2) We want to mark a point Q in the unit circle sch that the tangent has a value of 0.

Remember that:

tan(x) = sin(x)/cos(x)

So if sin(x) = 0, then:

tan(x) = sin(x)/cos(x) = 0/cos(x) = 0

So tan(x) is 0 in the points such that the sine function is zero.

These values are:

sin(0°) = 0

sin(180°) = 0

Then the two possible points where the tangent is zero are the ones drawn in the image below.