The point P (3,-2) lies on the terminal arm of an angle theta in standard position.
A) Determine the exact values of the six trigonometric ratios
B) Determine the value of principal angle and the related acute angle
To find the trigonometric ratios of angle theta, we can use the coordinates of point P:
x = 3
y = -2
First, we can find the hypotenuse of the right triangle formed by the point P and the origin (0,0) using the Pythagorean theorem:
r = sqrt(x^2 + y^2)
r = sqrt(3^2 + (-2)^2)
r = sqrt(9 + 4)
r = sqrt(13)
Now we can find the trigonometric ratios:
A) Trigonometric Ratios:
sin(theta) = y / r = -2 / sqrt(13)
cos(theta) = x / r = 3 / sqrt(13)
tan(theta) = y / x = -2 / 3
csc(theta) = 1 / sin(theta) = sqrt(13) / -2
sec(theta) = 1 / cos(theta) = sqrt(13) / 3
cot(theta) = 1 / tan(theta) = 3 / -2
B) Principal angle and related acute angle:
Since point P is in the quadrant IV (x > 0, y < 0), the principal angle is found by finding the angle whose terminal arm lies in the quadrant IV and has the same value as theta. This can be found using the inverse trigonometric functions:
principal angle = atan(-2 / 3)
To find the related acute angle, we subtract the principal angle from 180 degrees:
related acute angle = 180 - atan(-2 / 3)