### Video Transcript

Write three sin 45 degrees plus
seven cos 45 degrees minus six tan 45 degrees in the form root 𝑎 plus 𝑏, where 𝑎
and 𝑏 are integers.

There are several trigonometric
values we need to know by heart and there’s a little trick we can use to help
us. We use this table and it will help
us work out the values for sin 𝜃, cos 𝜃, and tan 𝜃 when 𝜃 is 30 degrees, 45
degrees, and 60 degrees. We begin by writing one, two,
three; three, two, one in the first two rows. We turn each of these numbers into
a fraction by adding a denominator of two. We then find the square root of the
numerator in each case. Remember though the square root of
one is simply one. So we don’t worry about writing it
on sin of a 30 and cos of 60.

There are no nice little tricks to
help us remember the values for tan 𝜃. We’re just going to need to learn
these. We can use this table now to work
out the values of sin of 45, cos of 45, and tan of 45 degrees as required in the
question.

Sin of 45 degrees we can see is
root two over two. So three sin of 45 degrees is three
multiplied by root two over two. And if we treat three as a fraction
with a denominator of one, we can simply multiply the numerators, three multiplied
by root two is three root two, and then the denominators, one multiplied by two is
two.

We repeat this process with seven
cos of 45 degrees. Cos 45 is root two over two. So we multiply this by seven. And once again, treating seven like
a fraction with the denominator of one, we get seven root two over two. Tan of 45 is a little easier. It’s one. So six tan 45 is six multiplied by
one which is of course six.

And we can now say that three sin
of 45 degrees plus seven cos of 45 degrees minus six tan of 45 degrees is three root
two over two plus seven root two over two minus six. The denominators of these two
fractions is the same. So we can simply add their
numerators. Three root two plus seven root two
becomes 10 root two. So our expression becomes 10 root
two over two minus six and 10 divided by two is five. So we have five root two minus
six.

We aren’t finished though. We were told to write our answer in
the form root 𝑎 plus 𝑏, where 𝑎 and 𝑏 are integers. That’s whole numbers. Our value for 𝑏 is negative
six. That is indeed a whole number, so
we’re done there. But we have five root two rather
than a single root of some whole number. We’re going to need to manipulate
five root two and we’re going to use this fact: the square root of 𝑎 multiplied by
the square root of 𝑏 is the square root of 𝑎 multiplied by 𝑏.

And since we know that five is the
square root of 25, we’re going to rewrite this a little bit as the square root of 25
multiplied by the square root of two. 25 multiplied by two is 50. So using that rule we just
mentioned, the square root of 25 multiplied by the square root of two is the square
root of 50. And we’re finished.

Three sin of 45 degrees plus seven
cos of 45 degrees minus six tan of 45 degrees can be written as root 50 minus
six. And we weren’t asked to specify our
values for 𝑎 and 𝑏. But we did already say that 𝑏 was
equal to negative six. So for completeness, we can say
that 𝑎 is equal to 50, which is of course an integer, a whole number. We’re done.