the point P(1/2, root 3/2) lies on the unit circle.
a) sketch the unit circle showing the point P and the standard position angle theta drawn to the terminal arm containing point P.
b) determine the exact values of the six trigonometric ratios of theta.
To sketch the unit circle, follow these steps:
a) Start by drawing a circle with a center at the origin (0, 0) and a radius of 1 unit.
b) Mark the x-axis and y-axis on the coordinate plane.
c) Plot the point P(1/2, √3/2) on the unit circle. To do this, locate the x-coordinate 1/2 on the x-axis and draw a vertical line to the point where it intersects the unit circle. Then locate the y-coordinate √3/2 on the y-axis and draw a horizontal line to the point where it intersects the unit circle. The intersection of these two lines will be point P.
To determine the exact values of the six trigonometric ratios of θ, we need to find the values of sine, cosine, tangent, cosecant, secant, and cotangent. Here's how:
Given: P(1/2, √3/2), we can calculate the trigonometric ratios using the coordinates of point P.
Let θ be the standard position angle.
a) Sine (sinθ) = y-coordinate / radius = √3/2 / 1 = √3/2
b) Cosine (cosθ) = x-coordinate / radius = 1/2 / 1 = 1/2
c) Tangent (tanθ) = sinθ / cosθ = (√3/2) / (1/2) = √3
d) Cosecant (cscθ) = 1 / sinθ = 1 / (√3/2) = 2 / √3 = (2√3) / 3
e) Secant (secθ) = 1 / cosθ = 1 / (1/2) = 2
f) Cotangent (cotθ) = 1 / tanθ = 1 / √3 = √3 / 3
Therefore, the exact values of the six trigonometric ratios of θ are:
sinθ = √3/2
cosθ = 1/2
tanθ = √3
cscθ = (2√3) / 3
secθ = 2
cotθ = √3 / 3
To answer part (a):
1. Start by drawing the coordinate axes (x-axis and y-axis) on a Cartesian plane. Label the x-axis and y-axis accordingly.
2. Since the point P(1/2, √3/2) lies on the unit circle, plot this point on the graph. To do this, count 1/2 unit on the x-axis from the origin and √3/2 units on the y-axis from the origin. Mark the point where these two values intersect. This is point P.
3. Next, draw the radius of the unit circle from the origin (0,0) to the point P(1/2, √3/2). This represents the distance from the origin to point P.
4. Finally, draw the angle θ in standard position, starting from the positive x-axis and moving counterclockwise until it intersects the radius at point P.
To answer part (b):
Now, let's determine the exact values of the six trigonometric ratios of θ for the point P(1/2, √3/2).
1. Sine (sin θ): It is equal to the y-coordinate of the point P divided by the radius. Therefore, sin θ = (√3/2) / 1 = √3/2.
2. Cosine (cos θ): It is equal to the x-coordinate of the point P divided by the radius. Therefore, cos θ = (1/2) / 1 = 1/2.
3. Tangent (tan θ): It is equal to the sine of θ divided by the cosine of θ. Therefore, tan θ = (√3/2) / (1/2) = √3.
4. Cosecant (csc θ): It is the reciprocal of sine. Therefore, csc θ = 1 / sin θ = 1 / (√3/2) = 2/√3 = (2√3) / 3.
5. Secant (sec θ): It is the reciprocal of cosine. Therefore, sec θ = 1 / cos θ = 1 / (1/2) = 2.
6. Cotangent (cot θ): It is the reciprocal of tangent. Therefore, cot θ = 1 / tan θ = 1 / √3 = √3/3.
These are the exact values of the six trigonometric ratios for θ in point P(1/2, √3/2) on the unit circle.