Find the sec of 300 degrees without using a calculator.
We will begin by recalling that the secant function is the reciprocal of the cosine function such that the sec of 𝜃 is equal to one over the cos of 𝜃. This means that the sec of 300 degrees is equal to one over the cos of 300 degrees. Next, we will sketch the unit circle in order to establish which quadrant 300 degrees lies in. We know that any angle in standard position is measured from the positive 𝑥-axis. If the angle is positive, as in this case, we measure in the counterclockwise direction.
Marking on the angles 90, 180, 270, and 360 degrees, we see that 300 degrees lies in the fourth quadrant. This is because 300 is greater than 270 but less than 360. We know that any point that lies on the unit circle has coordinates cos 𝜃, sin 𝜃. This means the point at which the terminal side of our angle intersects the unit circle has coordinates cos of 300 degrees, sin of 300 degrees. In the fourth quadrant, the 𝑥-coordinate will be positive, and the 𝑦-coordinate is negative. This means that the cos of 300 degrees must be positive.
In order to calculate the value of the cos of 300 degrees, we will begin by marking on the reference angle 𝛼. This is the measure of the acute angle between the terminal side and the 𝑥-axis in our diagram. Since the angles in a full turn or revolution sum to 360 degrees, we have 𝛼 plus 300 degrees is equal to 360 degrees. Subtracting 300 degrees from both sides of this equation gives us 𝛼 is equal to 60 degrees. The reference angle in this question is equal to 60 degrees.
Next, we can draw a perpendicular line from the 𝑥-axis to the point of intersection as shown. This creates a right triangle with a hypotenuse of length one. The side adjacent to the 60-degree angle and the right angle has length equal to the absolute value of the cos of 300 degrees. And since the cos of 300 degrees is positive, this is the length of the adjacent side.
We can now use our knowledge of right angle trigonometry, where the cos of angle 𝜃 is equal to the adjacent over the hypotenuse. This means that the cos of 60 degrees is equal to the cos of 300 degrees over one. 60 degrees is one of our special angles. And we know that the cos of this angle is equal to one-half. The cos of 300 degrees is therefore also equal to one-half. We can now use this to calculate the value of the sec of 300 degrees. This is equal to one divided by a half, which equals two.
Using our knowledge of trigonometric identities, reference angles, and the unit circle, we have found that the sec of 300 degrees is two.