Arnav K. answered • 10/24/20
Experienced in Calculus
Let's start with the givens. First, we know that cos(Θ) is greater than 0, which means that the cosine of the angle is positive. This can only occur in two quadrants, quadrant 1 and quadrant 4.
Next, we are told that tan(Θ) is 24/7. This value is positive, and the tangent of any angle can only be positive in two quadrants, quadrant 1 and quadrant 3.
Therefore, our angle must lie in quadrant 1, since we know that only in this quadrant will both given conditions be satisifed.
This is important because from here, we can draw the triangle formed by our angle. Because our angle is in the first quadrant, this means that both the x and y component of the triangle are positive. From the given fact that tan(Θ) is 24/7 (remember tangent is opposite over adjacent), we now that our triangle has a length of 7 and a height of 24.
From here, if we're going to want to solve for csc(Θ) and sec(Θ), we will need to solve for the hypotenuse of the triangle. Using Pythagorean Theorem:
sqrt(7^2 + 24^2)
sqrt(625)
= 25
We can now solve for the two trig identities. Let's start with csc(Θ). Remember, csc(Θ) is represented as hypotenuse over opposite. So this would be our hypotenuse, 25, divided by side that is opposite to the angle, which is 24. So:
csc(Θ) = 25/24
Finally, sec(Θ) is represented as hypotenuse over adjacent. This would be the hypotenuse, 25, divided by the adjacent side to the angle, which is 7. So:
sec(Θ) = 25/7
I hope this helps! It's a bit tricky without being able to draw figures, but if you need clarification let me know!
