### Video Transcript

By squaring both sides or
otherwise, solve the equation. Four sin π minus four cos π is
equal to root three, where π is greater than zero but less than or equal to 360
degrees. Be careful to remove any extraneous
solutions. Give your answers to two decimal
places.

Whilst the question suggests
squaring both sides, it will be easier to use one of our trigonometrical
identities. One of our addition formulae states
that sin of π minus πΌ is equal to sin π multiplied by cos πΌ minus cos π
multiplied by sin πΌ. Using this identity allows us to
write any equation in the form π sin π minus π cos π in the form π multiplied
by sin of π minus πΌ. The value of π is equal to the
square root of π squared plus π squared, where π and π are the coefficients of
sin π and cos π. And the angle πΌ is equal to the
inverse tan of π over π.

In our equation four sin π minus
four cos π, then π is equal to four and π is also equal to four. This means that π is equal to the
square root of four squared plus four squared. Four squared is equal to 16. Therefore, π is equal to the
square root of 16 plus 16. This simplifies to π is equal to
root 32. Root 32 can be rewritten as root 16
multiplied by root two. As the square root of 16 is equal
to four, π is equal to four root two. The angle πΌ is equal to inverse
tan or tan to the minus one of four divided by four. Four divided by four is equal to
one. The inverse tan of one is equal to
45 degrees, as tan of 45 equals one.

This means that the expression four
sin π minus four cos π can be rewritten as four root two sin of π minus 45. As four sin π minus four cos π
was equal to root three, four root two sin π minus 45 will also be equal to root
three. In order to calculate our values
for π, we firstly need to divide both sides of the equation by four root two. On the left-hand side, this leaves
us with sin of π minus 45. On the right-hand side, weβre left
with root six over eight or 0.3061 as a decimal.

Our next step is to take the
inverse of sin or sin to the minus one of both sides. This gives us π minus 45 is equal
to sin to the minus one of 0.3061, etcetera. Typing this into the calculator
gives us an answer of 17.83 to 2 decimal places. Adding 45 to this will give us one
of our solutions. However, we want all the solutions
between zero and 360 degrees. One way of doing this is by
considering the sin graph between zero and 360 degrees. We know that this is wave-shaped
and is symmetrical. We can see from the graph that the
sin of 17.83 is equal to the sin of 162.17, as theyβre both at the same height on
the π¦-axis.

This means that the inverse sin of
0.3061, etcetera, is also equal to 162.17. An alternative method to find the
second solution would be to remember one of our rules. This is that the second solution
will be 180 degrees minus the first solution. This is also sometimes known as the
cast method. If the sin of π is equal to a
positive value, thereβll be one solution between zero and 90 degrees and another
solution between 90 on 180 degrees. Our final step in this question is
to add 45 to both of our solutions. 17.83 plus 45 is equal to 62.83 and
162.17 plus 45 is equal to 207.17. Our two solutions for π are 62.83
degrees and 207.17 degrees. Both of these have been given to
two decimal places.