tahayassen said:
The value of tan2x can be calculated using the following formula: tan2x = 2tanx/(1-tan^2x) where x is the given angle. In this case, x is within the range of [3pi/2, 2pi], so we can use the reference angle of pi/2 and the quadrant principle to determine the sign of tanx. Since cosx is positive (12/13), we know that x is in the fourth quadrant where both sinx and cosx are negative. Therefore, tanx = -5/12. Plugging this into the formula, we get tan2x = 2(-5/12)/(1-(-5/12)^2) = -120/119.
The given range is [3pi/2, 2pi], which is equivalent to the fourth quadrant on the unit circle. Therefore, x can take on any value between 3pi/2 and 2pi, including the endpoints.
The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. In this problem, the reference angle is pi/2 since it forms a right triangle with the given cosx value of 12/13.
The quadrant principle states that in the first quadrant, all trigonometric functions are positive; in the second quadrant, only sinx is positive; in the third quadrant, only tanx is positive; and in the fourth quadrant, only cosx is positive. In this problem, cosx is positive (12/13), so we know that x is in the fourth quadrant where both sinx and cosx are negative, therefore tanx is also negative.
The value of tan2x can be used to find the value of x by using the inverse tangent function (arctan). In this case, x = arctan(-120/119) = 2.95 radians or approximately 168.9 degrees.