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
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.