Video Transcript
Given that the csc of ๐ is negative seven over six and tan of ๐ is greater than zero, find the cos of ๐.
Letโs think about the information weโre given. We know that the csc of this angle is negative seven over six and we know that the tan is positive. We want to remember the six trig functions. Since we were given the csc of ๐, we were given the hypotenuse over the opposite. And since weโre looking for the cos of this angle, weโre looking for the adjacent side length over the hypotenuse. But thereโs one other bit of information we need to consider. And that is the location of this angle on a coordinate grid.
We know that the tangent is positive. And the tan of an angle is only positive in the first and third quadrants. But we also know that the cosecant is negative. And if the cosecant is negative, sine will be negative. In the first quadrant, all three of the relationships are positive. In the third quadrant, the tangent is positive, but both the sine and cosine relationships are negative. And that means we know that our relationship, our cosine relationship, must be negative.
So far, we know the hypotenuse is seven and the opposite side length is six. In order to solve our problem, we need to know the adjacent side length. If we think about this in terms of a right-angled triangle, the measures of the side lengths must be positive because distance is always measured with positive values. And that means weโll need to use the absolute value for the hypotenuse. Instead of negative seven, it will be positive seven. And the opposite side length will be six.
We know we could use the Pythagorean theorem to solve for that third missing side length, the adjacent side length. So we say six squared plus ๐ squared equals seven squared. 36 plus ๐ squared equals 49. Subtracting 36 from each side, we get ๐ squared equals 13. From there, we take the square root of both sides to find our missing side length to be the square root of 13. Our cosine relationship is the adjacent side length, the square root of 13, over the hypotenuse, which we already know is seven. And because this falls in the third quadrant, this cosine relationship must be negative. And that makes our final answer negative square root of 13 over seven.
