Final answer:
To find <0, use the trigonometric identities of secant and tangent. sec(<0) = -13/5 and tan(<0) is negative. Use these identities to find the values of sin(<0), cos(<0), and tan(<0). Finally, determine the quadrant in which <0 lies and provide the value.
Step-by-step explanation:
To find <0 with the given information, we can use the trigonometric identities of secant and tangent. We know that sec(<0) = -13/5 and tan(<0) is negative.
First, we can use the identity sec(<0) = 1/cos(<0) to find the value of cos(<0). We know that sec(<0) = -13/5, so cos(<0) = 5/-13 = -5/13.
Next, we can use the identity sec^2(<0) = 1 + tan^2(<0) to find the value of tan(<0). We know that sec^2(<0) = (-13/5)^2 = 169/25, so tan^2(<0) = 169/25 - 1 = 144/25. Since tan(<0) is negative, tan(<0) = -12/5.
Now, we can use the Pythagorean identity sin^2(<0) + cos^2(<0) = 1 to find the value of sin(<0). We know that cos(<0) = -5/13, so sin^2(<0) = 1 - (-5/13)^2 = 1 - 25/169 = 144/169. Taking the square root of both sides, sin(<0) = ±12/13.
Since tan(<0) is negative and sin(<0) can be positive or negative, we can conclude that <0 is in either the second or fourth quadrant. In the second quadrant, sin(<0) is positive, but tan(<0) is negative, so it is not a valid solution. In the fourth quadrant, sin(<0) is negative and tan(<0) is negative, so it is a valid solution.
Therefore, <0 is in the fourth quadrant and its value is -12/13.