Video: Using Periodic, Odd, or Even Identities to Evaluate a Trigonometric Function Involving Special Angles

Find sec (−30°) without using a calculator.

03:38

Video Transcript

Find sec of minus 30 degrees without using a calculator.

Let us start by recording what sec actually means. We have that sec of 𝜃 is equal to one over cos 𝜃. So therefore, we can write sec of minus 30 is equal to one over cos of minus 30. Now we just need to find the value of cos of minus 30. Now since minus 30 is not a nice angle to work with, we can look at a cos graph to try and find another angle 𝜃 for which cos of 𝜃 is equal to cos of minus 30.

So let’s draw a cos graph. Here we have our cos graph. And we can mark on the angle minus 30. The horizontal dashed line represents the line 𝑦 is equal to cos of minus 30. Therefore, any other point at which this line intersects the line 𝑦 is equal to cos 𝜃 will give us another value of 𝜃, for which cos of minus 30 is equal to cos 𝜃. So let’s find another value of 𝜃. We can see that it also intersects at a point between naught and 90 degrees. Since the graph of cos is symmetric about the 𝑦-axis, we know that this point is at 30 degrees. And from this we can write cos of minus 30 is equal to cos of 30.

And now we can substitute this back into our original equation. And this gives us that sec of minus 30 is equal to one over cos of 30. So all that remains to do is to find the value of cos 30. In order to do this, let’s start by drawing an equilateral triangle with sides of length two.

So here is our equilateral triangle. Now, we can split it in half down the middle. Now, if we consider the triangle on the left, we can see that we have a right triangle with angles of 30 degrees and sixty degrees. Let’s draw this triangle out separately.

So this triangle has a hypotenuse of two and one of the sides of length one. And now we can use Pythagoras to find the other side. And this will give us a side of length root three. Next, we will use SOHCAHTOA in order to find the value of cos of 30 degrees. Now, since we’re trying to find cos of an angle, we’re interested in CAH. This tells us that cos of an angle 𝜃 is equal to the adjacent over the hypotenuse. Now, the angle we’re interested in is 30 degrees. So let’s call 30 degrees 𝜃. And we can label on the adjacent, opposite, and hypotenuse of the triangle.

So the adjacent is the side of length root three. The opposite is the side of length one. And the hypotenuse is the side of length two. Now we can write that cos of 30 is equal to the adjacent over the hypotenuse. So that’s simply root three over two. And now we can take this value of cos 30 and put it back into our equation in order to find a value of sec of minus 30. We get that sec of minus 30 is equal to one over root three over two.

This can be simplified to give us the solution that sec of minus 30 is equal to two over root three.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.