if csc theta= -5/3 and theta has its terminal side in quadrant III, find the exact value of tan 2 theta.
csc(θ)=-5/3
1/sin(θ)=-5/3
sin(θ)=1/(-5/3)=-3/5
cos(θ)=-√(1-(-3/5)²)
=-4/5
tan(θ)=sin(θ)/cos(θ)
=(-3/5)/(-4/5)
=3/4
tan(2θ)=2tan(&theta))/(1-tan²(θ))
=2(3/4)/(1-(3/4)²)
=3.43
(note: 2θ is in the first quadrant.
To find the exact value of tan(2θ), we need to use double angle identities in terms of trigonometric functions.
Given that csc(θ) = -5/3, we know that sine is the reciprocal of csc. Therefore, sine of θ is -3/5.
Since the terminal side of θ is in Quadrant III, cosine is negative. We can find cosine using the Pythagorean identity:
cos^2(θ) + sin^2(θ) = 1
cos^2(θ) + (-3/5)^2 = 1
cos^2(θ) + 9/25 = 1
cos^2(θ) = 16/25
cos(θ) = -4/5
Now we can find tan(θ) using the definitions:
tan(θ) = sin(θ)/cos(θ)
tan(θ) = (-3/5) / (-4/5)
tan(θ) = 3/4
Using the double angle identity for tangent, we have:
tan(2θ) = (2 * tan(θ)) / (1 - tan^2(θ))
tan(2θ) = (2 * (3/4)) / (1 - (3/4)^2)
tan(2θ) = (6/4) / (1 - (9/16))
tan(2θ) = (3/2) / (7/16)
tan(2θ) = (3/2) * (16/7)
tan(2θ) = 48/14
tan(2θ) = 24/7
Therefore, the exact value of tan(2θ) is 24/7.
To find the exact value of tan 2theta, we need to first find the values of sine and cosine of theta.
We are given that csc theta = -5/3. The reciprocal of csc theta is sin theta.
Reciprocal of -5/3 = -3/5
Therefore, sin theta = -3/5.
Now, since the terminal side of theta is in Quadrant III, both sine and cosine will be negative in this quadrant.
To find cosine of theta, we can use the Pythagorean Identity: sin^2(theta) + cos^2(theta) = 1.
Substituting the value of sin theta, we have (-3/5)^2 + cos^2(theta) = 1.
Simplifying, 9/25 + cos^2(theta) = 1.
cos^2(theta) = 1 - 9/25.
cos^2(theta) = 16/25.
Taking square root on both sides, we get cos theta = -4/5.
Now, we can use the identity tan 2theta = 2tan(theta) / (1 - tan^2(theta)).
Substituting the values we found, tan 2theta = 2(tan(theta)) / (1 - tan^2(theta)).
Since we know sin theta and cos theta, we can use the ratio tan theta = sin theta / cos theta.
tan theta = (-3/5) / (-4/5) = 3/4.
Now, substitute this value in the formula.
tan 2theta = 2(3/4) / (1 - (3/4)^2).
Simplifying further, tan 2theta = 6/4 / (1 - 9/16).
tan 2theta = 3/2 / (7/16).
To divide by a fraction, we can multiply by its reciprocal.
tan 2theta = 3/2 * (16/7).
tan 2theta = 48/14.
To simplify it further, we can divide both numerator and denominator by their common factor, which is 2.
tan 2theta = 24/7.
Therefore, the exact value of tan 2theta is 24/7.