If cos of theta= -7/12 and theta is in quadrant 2 then find
a. tan(theta) cot (theta)=
b. csc(theta) tan(theta)=
c. sin^2(theta)+ cos^2(theta)=
cosθ = x/r, so
x = -7
r = 12
y = √95
Now, recall that
sinθ = y/r
tanθ = y/x
and let 'er rip.
(a) and (c) should require no calculation at all...
To find the values of trigonometric functions, given the value of another trigonometric function, follow these steps:
Step 1: Determine the values of sin(theta) and cos(theta) using the given information.
Given: cos(theta) = -7/12
Using the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1
Substituting the value of cos(theta) into the equation, we can solve for sin(theta):
sin^2(theta) + (-7/12)^2 = 1
sin^2(theta) + 49/144 = 1
sin^2(theta) = 1 - 49/144
sin^2(theta) = 95/144
Step 2: Determine the sign of sin(theta) based on the quadrant information.
The given information states that theta is in Quadrant 2. In Quadrant 2, sin(theta) is positive.
So, sin(theta) = √(95/144) (Taking the positive square root since sin(theta) is positive)
Now, we can find the values of the trigonometric functions.
a. tan(theta) = sin(theta) / cos(theta)
tan(theta) = (√(95/144)) / (-7/12)
tan(theta) = -(12√95) / 7
cot(theta) = 1 / tan(theta)
cot(theta) = -7 / (12√95)
b. csc(theta) = 1 / sin(theta)
csc(theta) = 1 / (√(95/144))
csc(theta) = 12 / √95
tan(theta) = sin(theta) / cos(theta)
tan(theta) = (√(95/144)) / (-7/12)
tan(theta) = -(12√95) / 7
csc(theta) * tan(theta) = (12 / √95) * -(12√95) / 7
csc(theta) * tan(theta) = -1440 / 665
c. The identity sin^2(theta) + cos^2(theta) = 1 is a fundamental trigonometric identity known as the Pythagorean identity. It holds true for all values of theta.
So, sin^2(theta) + cos^2(theta) = 1