Determine the integral of eight cos
five 𝜃 d𝜃 between negative 𝜋 by four and negative 𝜋.
Our first step here is to take out
the constant eight. This leaves us with eight
multiplied by the integral of cos five 𝜃 d𝜃. The integral of cos 𝜃 is a
standard one that we should know. It is equal to sin 𝜃, as
integrating is the opposite, or inverse, of differentiating.
We want to integrate cos five
𝜃. This means we need to use another
general rule. This states that the integral of
cos 𝑛𝜃 is equal to one over 𝑛 multiplied by sin 𝜃. We differentiate the 𝑛𝜃 to give
us 𝑛. The integral of cos five 𝜃 is
equal to one-fifth multiplied by sin five 𝜃. We need to multiply this by eight
and have limits of negative 𝜋 by four and negative 𝜋. As with the eight at the start, we
can take the constant one-fifth outside of the bracket. This gives us eight-fifths
multiplied by sin five 𝜃.
Our next step is to substitute in
our two limits and subtract the answers. Substituting in the upper limit
gives us sin of five multiplied by negative 𝜋 over four. This can be rewritten as sin of
negative five 𝜋 over four. Substituting in our lower limit
gives us sin of five multiplied by negative 𝜋. Once again, this can be rewritten
as sin of negative five 𝜋.
At this point, it is worth drawing
the sine curve to see if negative five 𝜋 by four and negative five 𝜋 correspond
with any our known angles. The sine curve has a maximum value
of one and a minimum value of negative one. It has key values on the 𝜃-, or
𝑥-axis, of 𝜋 by two and 𝜋, negative 𝜋 by two and negative 𝜋. If you prefer to think of these
angles in degrees. It’s worth remembering that 𝜋
radians is equal to 180 degrees.
The sine curve looks as shown in
the diagram. However, at the moment, we have a
slight problem as our two angles negative five 𝜋 by four and negative five 𝜋 don’t
fit in the range. As the sine curve has a period of
two 𝜋, it repeats every two 𝜋 radians, we can continue the graph as shown.
We can see clearly from the graph
that the sin of negative five 𝜋 is equal to zero. Negative five 𝜋 by four is shown
in the diagram. By going vertically upwards to the
sine curve and then horizontally along to the 𝑦-axis, we can see what this value
will take. Due to the symmetry of the sine
curse, sin of negative five 𝜋 by four is equal to sin of 𝜋 by four.
𝜋 by four is equal to 45 degrees
and is one of our known angles. This is equal to root two over
two. The sin of 45 degrees equals root
two over two. Therefore, the sin of 𝜋 by four
radians must also equal root two over two. The sin of negative five 𝜋 by four
is equal to root two over two. And the sin of negative five 𝜋 is
equal to zero.
Root two over two minus zero is
root two over two. So, we need to multiply this by
eight-fifths. Multiplying the numerators gives us
eight root two. And multiplying the denominators,
gives us 10. We have eight root two over 10. Eight and 10 have a common factor
of two. So, we can divide the numerator and
denominator by two. This gives us four root two over
five. We can, therefore, say that the
integral of eight cos five 𝜃 d𝜃 between negative 𝜋 by four and negative 𝜋 is
equal to four root two over five.