The terminal side of an angle in standard position goes through the poin (-2,5). Find the values of the 6 trig function
sin
cos
tan
csc
sec
cot
did you make a diagram?
you would have a right-angled triangle in the second quadrant with x=-2 and r=5
so
x^2 + y^2 = r^2
4 + y^2 = 25
y = ± √21
but we are in quadrant II, so y = +√21
now, knowing the trig rations in terms of x,y, and r, your question is easy
e.g. tan(angle) = y/x = - √21/2
sec(angle) = r/x = - 5/2
you do the rest.
To find the values of the six trigonometric functions (sin, cos, tan, csc, sec, cot) for an angle in standard position, you need to know the coordinates of the point where the terminal side of the angle intersects the unit circle.
Here's how you can find these values:
1. Determine the angle formed by the terminal side and the positive x-axis. To do this, you can use the coordinates of the given point (-2,5). Since the point lies in the second quadrant (negative x-coordinate, positive y-coordinate), the angle formed is between 90 degrees and 180 degrees.
2. Find the radius (or hypotenuse) of the unit circle. Since the unit circle has a radius of 1, the value will be 1.
3. Determine the lengths of the legs (opposite and adjacent) of the right triangle formed by the terminal side, x-axis, and a perpendicular line from the terminal side to the x-axis. Using the coordinates (-2,5), the opposite leg is 5 units and the adjacent leg is -2 units (since it is in the second quadrant).
4. Calculate the trigonometric function values using the determined angle and the lengths of the triangle's sides:
a. Sin: sin(theta) = opposite/hypotenuse = 5/1 = 5
b. Cos: cos(theta) = adjacent/hypotenuse = -2/1 = -2
c. Tan: tan(theta) = opposite/adjacent = 5/-2 = -2.5
d. Csc: csc(theta) = 1/sin(theta) = 1/5
e. Sec: sec(theta) = 1/cos(theta) = 1/-2 = -0.5
f. Cot: cot(theta) = 1/tan(theta) = 1/-2.5 = -0.4
So, the values of the six trigonometric functions are as follows:
sin(theta) = 5
cos(theta) = -2
tan(theta) = -2.5
csc(theta) = 1/5
sec(theta) = -0.5
cot(theta) = -0.4