Find all the trigonometric ratios of 𝜃 given the cot of 𝜃 is equal to negative eight over 15 where 𝜃 is between three 𝜋 over two and two 𝜋.
First, let’s consider the trigonometric ratios. We have sine, cosine, tangent, where sine is the opposite side length over the hypotenuse, cosine is the adjacent side length over the hypotenuse, tangent is the opposite over adjacent. We also have the inverse of the first three identities. Cosecant is the inverse of sine, and it is the hypotenuse over the opposite side length. Secant is the inverse of [cosine]; it’s the hypotenuse over the adjacent side length. And cotangent is the inverse of tangent and is the adjacent side length over the opposite side length.
We’re given one of the ratios, the cot of 𝜃, and we’ll need to find the remaining five. Since we know that the cot of 𝜃 is negative eight over 15, we can say that the tan of 𝜃 will be the inverse of that, negative 15 over eight. But we’ll have to do a bit more than that to find the other four. But first we’ll need to think about what we know about 𝜃. We know that 𝜃 falls between three 𝜋 over two and two 𝜋. If we think about a unit circle that starts at zero, 90 degrees is 𝜋 over two, 180 degrees is 𝜋, 270 degrees is three 𝜋 over two, and 360 degrees is two 𝜋.
Since our angle falls between three 𝜋 over two and two 𝜋, we’re dealing with an angle that falls in the fourth quadrant. And this gives us really important information. In the fourth quadrant, you have negative sine ratios, positive cosine ratios, and negative tangent ratios. This means we know that the sine ratio and the cosecant ratio will be negative, but the cosine and secant ratios will be positive. However, we’re still missing a crucial piece of information, a third side length.
If we let this angle be 𝜃, the opposite side length is 15, and the adjacent side length is eight. And we can use the Pythagorean theorem to find the missing third side, the hypotenuse. The Pythagorean theorem is 𝑐 squared equals 𝑎 squared plus 𝑏 squared, where 𝑐 is the hypotenuse of a right triangle, and 𝑎 and 𝑏 are the other two side lengths.
For us, 𝐻 squared is equal to eight squared plus 15 squared. 64 plus 225 equals 289, which is the hypotenuse squared. To find the hypotenuse, we need to take the square root of both sides of the equation. The square root of the hypotenuse squared is the hypotenuse. And the square root of 289 is plus or minus 17. We’ll label the hypotenuse 17.
Sin of our angle measure is equal to the opposite side length over the hypotenuse, 15 over 17. But because we know where its location is, we can say that sin of 𝜃 is negative 15 over 17. Cosecant is the inverse of that value, which is negative 17 over 15. From there, we’ll look for the cos of 𝜃, which is the adjacent side length over the hypotenuse, eight over 17. And that’s going to be a positive value. The secant of our angle measure will be the inverse of the cosine, so 17 over eight. For this angle, the sin is negative fifteen seventeenths. The cos is positive eight seventeenths. The tan, negative fifteen-eighths. The cosec, negative 17 over 15. The sec, 17 over eight. And the cot that we we’re given, negative eight over 15.