Given that csc θ = -4 and tan θ > 0, find the exact value of
a. x
b. sin θ
c. cos θ
d. r
e. sec θ
f. cot θ
same as the others ....
cscØ = -4
so sinØ = -1/4
we also know that tanØ>0 so
the sine is negative and the tangent is positive ---> Ø must be in III
since sinØ = -1/4, in III, y = -1, r = 4
by Pythagoras and the CAST rule
x = -√15
The rest is now easy, just use your definitions.
let me know what you get.
e.g.
cotØ = x/y = -√15/-1 = √15
To find the exact value of each trigonometric function, we need to determine the value of θ first. From the given information, we know that csc θ = -4 and tan θ > 0.
a) To find the value of x, we need to identify the triangle in which θ exists. Since csc θ = -4, we can use the Cosecant function in terms of the right triangle. The definition of csc θ is equal to the hypotenuse divided by the opposite side. Thus, we have:
csc θ = -4
⇒ 1/sin θ = -4
⇒ sin θ = -1/4
Since sin θ is negative and tan θ is positive, we conclude that θ lies in the third quadrant of the unit circle. Therefore, x must also be negative.
b) To find sin θ, we use the information sin θ = -1/4, which we obtained above.
c) To find cos θ, we can utilize the Pythagorean Identity, which states that sin^2 θ + cos^2 θ = 1. Since we know sin θ, we can solve for cos θ:
sin^2 θ + cos^2 θ = 1
(-1/4)^2 + cos^2 θ = 1
1/16 + cos^2 θ = 1
cos^2 θ = 1 - 1/16
cos^2 θ = 15/16
cos θ = ± √(15/16)
Since we are in the third quadrant, where cosine is negative, cos θ must be negative.
d) To find r, we use the Pythagorean Theorem:
r = √(x^2 + y^2)
r = √(x^2 + (-1/4)^2)
e) To find sec θ, we will use the reciprocal identity: sec θ = 1/cos θ. We can substitute the value we found for cos θ into this equation to obtain the value of sec θ.
f) To find cot θ, we will use the reciprocal identity: cot θ = 1/tan θ. Since we are given that tan θ > 0, we can substitute the value into this equation to find the value of cot θ.