Video Transcript
Find the exact value of three cos
75 minus three cos 15.
We could calculate the value of
this expression by simply typing it into our calculator. However, as we are asked to find
the exact value, we will use our knowledge of the special angles and their
trigonometric values. We recall that the special angles
are considered to be zero, 30, 45, 60, and 90 degrees. One way of recalling the sine and
cosine of these angles is as follows. We write the numbers zero, one,
two, three, and four in the sine row and write them in reverse order in the cosine
row.
Next, we divide each of the numbers
by four and, where possible, simplify. Zero divided by four is zero, two
divided by four simplifies to one-half, and four divided by four is equal to
one. We then take the square root of
each of our values. The square root of zero is
zero. Therefore, the sin of zero degrees
and cos of 90 degrees are equal to zero. The square root of one is one. Therefore, the sin of 90 degrees
and cos of zero degrees are equal to one. When square rooting a fraction, we
can square root the numerator and denominator separately. This means that the square root of
one-quarter is the same as the square root of one over the square root of four. This is equal to one-half. So the sin of 30 degrees and cos of
60 degrees is equal to one-half. The sin of 45 degrees and the cos
of 45 degrees are both equal to one over root two. By rationalizing the denominator,
we can also write this as root two over two. Finally, the sin of 60 degrees and
cos of 30 degrees are equal to root three over two.
Let’s now consider how this helps
us solve the problem in this question. We have the cos of 75 degrees and
the cos of 15 degrees. We notice that 75 is equal to 45
plus 30 and 15 is equal to 45 minus 30. This means that we can rewrite the
expression as three multiplied by cos of 45 plus 30 minus three multiplied by cos of
45 minus 30. We can then take out a factor of
three as shown.
Next, we recall the trigonometric
angle sum and difference identities. We know that the cos of 𝛼 plus 𝛽
is equal to cos 𝛼 cos 𝛽 minus sin 𝛼 sin 𝛽. This means that cos 45 plus 30 is
equal to cos 45 multiplied by cos 30 minus sin 45 multiplied by sin 30. We also know that the cos of 𝛼
minus 𝛽 is equal to cos 𝛼 cos 𝛽 plus sin 𝛼 sin 𝛽. The cos of 45 minus 30 is therefore
equal to cos 45 multiplied by cos 30 plus sin 45 multiplied by sin 30.
If we call these equations one and
two, we can substitute the right-hand sides into our expression. When we subtract equation two from
equation one, cos 45 cos 30 will cancel. Negative sin 45 sin 30 minus sin 45
sin 30 gives us negative two sin 45 sin 30. Our expression therefore simplifies
to negative six sin 45 sin 30. We see from our table that the sin
of 45 degrees is root two over two and the sin of 30 degrees is one-half. Substituting these into our
expression, we have negative six multiplied by root two over two multiplied by
one-half. We can divide the numerator and
denominator by two. This leaves us with negative three
root two over two. This is the exact value of three
cos 75 degrees minus three cos 15 degrees.
