Video Transcript
If 𝑥 is greater than or equal to
zero degrees and less than or equal to 360 degrees, then the number of solutions of
the equation four sin 𝑥 is equal to the tan of 𝑥 is what.
This equation involves two trig
ratios: sine and tangent. We recall that we can express the
tangent function in terms of the sine and cosine functions. The tan of 𝑥 is equal to the sin
of 𝑥 over the cos of 𝑥. Substituting this into the
right-hand side of our equation, we have four sin 𝑥 is equal to sin 𝑥 over cos
𝑥. Next, we can subtract sin 𝑥 over
cos 𝑥 from both sides, giving us four sin 𝑥 minus sin 𝑥 over cos 𝑥 equals
zero.
At this stage, we might be tempted
to divide the equation through by the shared factor of sin 𝑥. However, doing this could
potentially result in the loss of some solutions if the factor we divide by is equal
to zero. This means that instead we will
factor sin 𝑥 out of the left-hand side of the equation. This gives us sin 𝑥 multiplied by
four minus one over cos 𝑥 is equal to zero.
We now have a product that is equal
to zero. And the only way a product can
equal zero is if at least one of the factors itself equals zero. This means that we need to solve
the two equations sin 𝑥 equals zero and four minus one over cos 𝑥 equals zero. Recalling the graph of the sine
function as shown, we see that sin 𝑥 is equal to zero three times in the interval
where 𝑥 is greater than or equal to zero and less than or equal to 360 degrees. These solutions are zero, 180, and
360 degrees. However, in this question, we are
only interested in the number of solutions. sin 𝑥 is equal to zero three times
between zero and 360 degrees inclusive.
Let’s now consider the second
equation, four minus one over cos 𝑥 equals zero. Multiplying through by cos of 𝑥
gives us four cos of 𝑥 minus one equals zero. We can then add one to both sides
such that four cos 𝑥 equals one and, finally, divide through by four such that the
cos of 𝑥 is equal to one-quarter.
Recalling the graph of the cosine
function and drawing a horizontal line across the graph at 𝑦 equals one-quarter, we
find that there are two values of 𝑥 in the interval 𝑥 is greater than or equal to
zero and less than or equal to 360 degrees for which the cos of 𝑥 equals
one-quarter. Whilst we could calculate these
exact values, we don’t need to in this question. However, it is clear that these are
not the same values for which sin 𝑥 equals zero, as one of the solutions lies
between zero and 90 degrees and the second solution lies between 270 and 360
degrees.
We can therefore conclude that
there are five solutions to the equation four sin 𝑥 equals tan 𝑥 between zero and
360 degrees inclusive. The correct answer is five.
