In this problem, it is given that tan(x)= 2/3 and sec(x) < 0. It is also given to us we are going to use Quadrant III and the values to start our problem. Before we start solving, it is better to decide first if the sign is going to be negative or positive based on what is given to us. It can only work in ONE quadrant and only ONE of these quadrants can work (in this case it is tangent or Quadrant III).
Reading from left to right, this is one way to find the missing pieces accordingly. When solving for cotangent, you can use the reciprocal identity. Now that you know what cotangent is, you can use the Pythagorean Identity. Plug in the value of cotangent, add, and take the square root. From there, you will find your value for cosecant. Because you know the value for cosecant, you can use it to find the value for sine. Thus, you can use the reciprocal identity. Remember, you must rationalize in order to get rid of the square root on the denominator. Now, if you look back to see what trig you are missing, you find that you still need to solve for secant and cosine. Since tangent was given to us, we can use the Pythagorean identity to find secant. Once we find the value for secant, we can use the reciprocal identity to find cosine.
We can use our knowledge of SOH CAH TOA to find/check our values. This picture demonstrates the ratio and steps to get the value. Remember, "r" does NOT equal to 1. If you look back at the problem, it is given to us that "r", the radius, is 2 radical 13.
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