Given that sin 𝜃 equals three-fifths and 𝜃 is greater than zero and less than 𝜋 over two, evaluate cos 𝜃, tan 𝜃, sec 𝜃, csc 𝜃, and cot 𝜃.
In this question, we are given the value of sin 𝜃 and are asked to evaluate the other five trigonometric functions of the same angle. We are told that 𝜃 lies between zero and 𝜋 over two. And since angles in standard position are measured in a counterclockwise direction from the positive 𝑥-axis, we know that 𝜃 lies in the first quadrant. We can sketch a right triangle in the first quadrant where sin 𝜃 equals three-fifths as shown, as sin 𝜃 is equal to the opposite over the hypotenuse.
This right triangle is a Pythagorean triple consisting of the three positive integers three, four, and five such that three squared plus four squared is equal to five squared. Next, we recall from our CAST diagram that in the first quadrant, all six trigonometric functions are positive. We know that in any right triangle, the cos of angle 𝜃 is equal to the adjacent over the hypotenuse. This means that in our diagram cos 𝜃 is equal to four-fifths. The tan of any angle 𝜃 in a right triangle is equal to the opposite over the adjacent. In this question, tan 𝜃 is therefore equal to three-quarters. We could also calculate this by using the identity tan 𝜃 is equal to sin 𝜃 over cos 𝜃. Dividing three-fifths by four-fifths gives us three-quarters.
Next, we will consider the reciprocal functions: cosecant, secant, and cotangent. csc 𝜃 is equal to one over sin 𝜃, sec 𝜃 is equal to one over cos 𝜃, and cot 𝜃 is equal to one over tan 𝜃. When finding the reciprocal of a fraction, we simply swap the numerator and denominator. This means that sec 𝜃 is equal to five-quarters. In the same way, one over three-fifths is equal to five-thirds. So csc 𝜃 is equal to five-thirds. Finally, as cot 𝜃 is equal to one over tan 𝜃, this is equal to four-thirds.
We now have the five values required. cos 𝜃 is equal to four-fifths. tan 𝜃 is equal to three-quarters. sec 𝜃 is equal to five-quarters. csc 𝜃 is equal to five-thirds. And cot 𝜃 is equal to four-thirds. It is worth noting that we could’ve found the values of csc 𝜃, sec 𝜃, and cot 𝜃 directly from our right triangle, where csc 𝜃 is equal to the hypotenuse over the opposite. sec 𝜃 is equal to the hypotenuse over the adjacent. And cot 𝜃 is equal to the adjacent over the opposite.