Answer:
Explanation:
To determine the exact values of sine, cosine, and tangent for the given point (-1, √3) in standard position, we need to find the corresponding angle θ.
Step 1: Identify the coordinates of the point.
In this case, the given point is (-1, √3), which means the x-coordinate is -1 and the y-coordinate is √3.
Step 2: Find the radius r.
The radius is the distance from the origin (0, 0) to the given point. Using the distance formula, we can calculate the radius:
r = √((-1)^2 + (√3)^2) = √(1 + 3) = √4 = 2
Step 3: Determine the quadrant of the angle.
Since the x-coordinate is negative and the y-coordinate is positive, the point (-1, √3) lies in the second quadrant.
Step 4: Calculate the angle θ.
To find the angle θ, we can use the inverse tangent function since we have the y-coordinate and the x-coordinate. However, we need to consider the quadrant in which the angle lies. Since the point is in the second quadrant, the angle will be greater than 90 degrees but less than 180 degrees.
θ = atan(√3/-1) = atan(-√3) = -60 degrees
Step 5: Determine the exact values of sin, cos, and tan.
Using the calculated angle θ, we can find the exact values of sine, cosine, and tangent.
sin(θ) = sin(-60 degrees) = -√3/2
cos(θ) = cos(-60 degrees) = -1/2
tan(θ) = tan(-60 degrees) = √3
Therefore, the exact values of sin, cos, and tan for the point (-1, √3) in standard position are:
sin = -√3/2
cos = -1/2
tan = √3