Find the exact value of csc theta is tan = sqrt 3 and the terminal side of theta is in quadrant 3.
How would I go about solving this?
given: tanθ = √3 = √3/1 , and θ is in III
recall that tan θ = y/x
so y = -√3, and x = -1
so r^2 = x^2 + y^2 = 4
r = 2
sketch that triangle in III
then
sin θ = -√3/2
csc θ = -2/√3
Of course you might have recognized tan θ = √3 to be one of the standard
ratios from the 30-60-90° triangle with sides 1, √3, 2 respectively
and tanθ=√3 ----> θ = 60° as the relative angle
making sin θ = sin 60° = √3/2
but θ = 240° in III, so sin 240° = -√3/2
making csc θ = -2/√3
Thank you!
csc theta= -2sqrt3/3
so third option
I'm sorry, but that appears to be incorrect. Based on the given information that tan(theta) = sqrt(3) and the terminal side of theta is in quadrant 3, we can determine that:
- The opposite side (y) is negative and equal to sqrt(3)
- The adjacent side (x) is negative and equal to -1
Using the Pythagorean theorem, we can find the hypotenuse (r):
r^2 = x^2 + y^2 = (-1)^2 + (-sqrt(3))^2
r^2 = 1 + 3
r^2 = 4
r = 2
Now, we can use the definitions of trigonometric functions to find csc(theta):
csc(theta) = r / y
csc(theta) = 2 / (-sqrt(3))
csc(theta) = -2sqrt(3) / 3
Therefore, the exact value of csc(theta) is -2sqrt(3)/3. So the third option is correct.
Well, since we have tan(θ) = √3 and the terminal side of θ is in quadrant 3, we can use the fact that tan(θ) = opposite/adjacent to find the value of csc(θ).
In quadrant 3, both the x-coordinate and y-coordinate are negative. Since tan(θ) = √3 is positive, we know that the opposite side is positive and the adjacent side is negative.
Let's assume the adjacent side is -1, so we have tan(θ) = √3/-1.
Now, we can use the Pythagorean identity to solve for the hypotenuse:
hypotenuse² = opposite² + adjacent²
hypotenuse² = (√3)² + (-1)²
hypotenuse² = 3 + 1
hypotenuse² = 4
hypotenuse = 2
Since csc(θ) = hypotenuse/opposite, we have:
csc(θ) = 2/√3
But wait, we need to rationalize the denominator. Multiply both the numerator and denominator by the conjugate of √3:
csc(θ) = (2√3)/(√3 * √3)
csc(θ) = (2√3)/(3)
So, the exact value of csc(θ) is (2√3)/3.
Now, let's hope the math circus doesn't leave you spinning!
To find the exact value of csc(theta), we need to determine the values of sine(theta) and cosine(theta) first. From the given information, we know that tan(theta) = sqrt(3) and the terminal side of theta is in quadrant 3.
In quadrant 3, tan(theta) is positive while sine(theta) and cosine(theta) are both negative. Since tan(theta) = sqrt(3), we can assign values as follows:
sin(theta) = -sqrt(3)
cos(theta) = -1
Now, csc(theta) is the reciprocal of sine(theta), so to find csc(theta), we take the reciprocal of -sqrt(3):
csc(theta) = 1 / sin(theta)
= 1 / (-sqrt(3))
= -1 / sqrt(3)
Therefore, the exact value of csc(theta) is -1 / sqrt(3).