Final answer:
To find csc(θ) and tan(θ), we use the given cos(θ) = -5/13 to determine the y-value in the right triangle via the Pythagorean theorem. In quadrant 3, both sine and tangent are negative. Hence, csc(θ) = -13/12 and tan(θ) = 12/5.
Step-by-step explanation:
The student is asking to find the exact values of csc(θ) and tan(θ) given that the cosine of an angle θ in quadrant 3 is -5/13. In a right triangle, if the cosine of an angle is given by the ratio of the adjacent side to the hypotenuse, then cos(θ) = adjacent/hypotenuse = -5/13. This tells us that the hypotenuse (denoted as 'r') is 13 and the adjacent side (x-value) is -5.
Since θ is in quadrant 3, both sine and tangent will be negative. We can find the opposite side (y-value) using the Pythagorean theorem: x² + y² = r². Here, (-5)² + y² = 13² leads to y² = 169 - 25, so y² = 144. Therefore, y = -12, because in quadrant 3, the y-value is also negative.
The sine of θ is opposite/hypotenuse, which is sin(θ) = -12/13. The cosecant is the reciprocal of the sine, so csc(θ) = -13/12. The tangent of θ is opposite/adjacent, which is tan(θ) = -12/-5 = 12/5. To sum up, the exact values are:
- csc(θ) = -13/12
- tan(θ) = 12/5