Final answer:
To find sec(θ) and sin(θ) given tan(θ) = -5/12 in the second quadrant, we use the right triangle definitions of the trigonometric functions. Sin(θ) = opposite/hypotenuse = -5/13 and sec(θ) = 1/cos(θ) = -13/12.
Step-by-step explanation:
Given that tan(θ) = -5/12 and θ is in quadrant II, we first recognize that the tangent function is negative in quadrant II, while sin(θ) and cos(θ) are positive and negative, respectively. To find sec(θ) and sin(θ), we can use the definitions of trigonometric functions in terms of sides of a right triangle. In a right triangle where tan(θ) = opposite/adjacent, if we consider the opposite to be -5 (since tan is negative) and the adjacent to be 12, using the Pythagorean theorem, the hypotenuse (r) is calculated as sqrt((-5)2 + 122) = sqrt(25 + 144) = sqrt(169) = 13.
Since θ is in the second quadrant, sin(θ) will be positive. Therefore, sin(θ) = opposite/hypotenuse = -5/13 (here, we take negative because in the second quadrant the y-value, or the opposite side, is positive). To find sec(θ), which is the reciprocal of the cosine, we first find cos(θ) = adjacent/hypotenuse = -12/13 (cosine is negative in the second quadrant). Therefore, sec(θ) = 1/cos(θ) = -13/12.