Tan theta 12/5 ,pi <theta<3pi/2
Find sin 2theta
Think of a 5,12,13 right triangle. The principle angle (to the -y axis) is arcsin 12/13
Theta is in the third quadrant, and therefore
theta = pi + arcsin 12/13
sin 2theta = 2 sin theta cos theta
= 2*(-12/13)(-5/13) = 120/169 = 0.71006
Check: theta = 4.31760 radians
2 theta = 8.6352 radians
sin (2 theta) = 0.71006
To find sin 2theta, we first need to find the value of 2theta.
Given that tan(theta) = 12/5 and pi < theta < 3pi/2, we can determine the value of theta using the inverse tangent function (tan^(-1)).
tan(theta) = 12/5
theta = tan^(-1)(12/5)
Using a calculator, we can find the value of theta to be approximately 68.2 degrees.
Now, to find sin 2theta, we need to double the value of theta and then find the sine of it.
2theta = 2 * 68.2 = 136.4 degrees
Finally, we can find sin 2theta by taking the sine of 136.4 degrees.
sin 2theta = sin(136.4)
Using a calculator, we find that sin(136.4) is approximately -0.855.
Therefore, sin 2theta is approximately -0.855.
To find sin 2theta, we can use the trigonometric identity: sin 2theta = 2sin theta * cos theta.
Given that tan theta = 12/5 and pi < theta < 3pi/2, we can first find the values of sin theta and cos theta.
To find sin theta, we can use the fact that tan theta = sin theta / cos theta. We can solve for sin theta as follows:
tan theta = 12/5
sin theta / cos theta = 12/5
sin theta = 12/5 * cos theta
Next, we look at the given range of theta (pi < theta < 3pi/2). This range corresponds to the third quadrant on the unit circle, where sin(theta) is negative.
Since we know that sin theta = 12/5 * cos theta, and sin(theta) is negative, it means that cos(theta) must also be negative in the third quadrant.
Now, let's use the Pythagorean identity to relate sin theta and cos theta.
sin^2 theta + cos^2 theta = 1
(12/5 * cos theta)^2 + cos^2 theta = 1
144/25 * cos^2 theta + cos^2 theta = 1
(144/25 + 1) * cos^2 theta = 1
(144/25 + 25/25) * cos^2 theta = 1
(169/25) * cos^2 theta = 1
cos^2 theta = 25/169
cos theta = ± √(25/169)
cos theta = ± 5/13
Since cos theta is negative in the third quadrant, we take cos theta = -5/13.
Finally, plugging in the values of sin theta and cos theta into the identity sin 2theta = 2sin theta * cos theta, we get:
sin 2theta = 2sin theta * cos theta
sin 2theta = 2 * (12/5 * cos theta) * cos theta
sin 2theta = (24/5) * (cos theta)^2
sin 2theta = (24/5) * (-(5/13))^2
sin 2theta = (24/5) * (25/169)
sin 2theta = -24/13
Therefore, sin 2theta is equal to -24/13.