Cos theta=9/41 and the terminal arm of the angle is in quad.4. Find remaining trigonometri functions?
Use trigonometric identities and reciprocals of sin cos and tan to find other trigonometric functions 😁
Good
To find the remaining trigonometric functions, we first need to determine the values of sine, cosine, and tangent.
Given that cos(theta) = 9/41 and the terminal arm is in quadrant 4, we can determine that sin(theta) is negative because in quadrant 4, only the y-coordinate is negative.
Step 1: Find sin(theta):
To find sin(theta), we can use the Pythagorean identity sin^2(theta) = 1 - cos^2(theta).
Since cos(theta) = 9/41, we can solve for sin(theta) as follows:
sin^2(theta) = 1 - (9/41)^2
sin^2(theta) = 1 - 81/1681
sin^2(theta) = (1681 - 81)/1681
sin^2(theta) = 1600/1681
Taking the square root of both sides, we get:
sin(theta) = sqrt(1600/1681)
sin(theta) = 40/41
Since the terminal arm is in quadrant 4 (where the x-coordinate is positive and the y-coordinate is negative), sin(theta) is negative:
sin(theta) = -40/41
Step 2: Find tan(theta):
To find tan(theta), we can use the formula tan(theta) = sin(theta) / cos(theta).
Since we have already found sin(theta) and cos(theta), we can substitute the values into the formula:
tan(theta) = (-40/41) / (9/41)
tan(theta) = -40/9
So, the remaining trigonometric functions for cos(theta) = 9/41 in quadrant 4 are:
sin(theta) = -40/41
tan(theta) = -40/9
Note: The values for cosec(theta), sec(theta), and cot(theta) can be found by taking the reciprocals of sin(theta), cos(theta), and tan(theta), respectively.
oops
tan(Θ) = y/x
cos(Θ) = 9/41 = x/r
use Pythagoras to find y
... x^2 + y^2 = r^2
sin(Θ) = y/r ... remember Q IV
tan(Θ) = y/r