Question Video: Solving a Trigonometric Equation Using Squaring

If 0Β° β©½ π‘₯ β©½ 360Β°, then the number of solutions of the equation 4 sin π‘₯ = tan π‘₯ is οΌΏ.

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Video Transcript

If π‘₯ 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 π‘₯ is equal to the tan of π‘₯ 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 π‘₯ is equal to the sin of π‘₯ over the cos of π‘₯. Substituting this into the right-hand side of our equation, we have four sin π‘₯ is equal to sin π‘₯ over cos π‘₯. Next, we can subtract sin π‘₯ over cos π‘₯ from both sides, giving us four sin π‘₯ minus sin π‘₯ over cos π‘₯ equals zero.

At this stage, we might be tempted to divide the equation through by the shared factor of sin π‘₯. 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 π‘₯ out of the left-hand side of the equation. This gives us sin π‘₯ multiplied by four minus one over cos π‘₯ 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 π‘₯ equals zero and four minus one over cos π‘₯ equals zero. Recalling the graph of the sine function as shown, we see that sin π‘₯ is equal to zero three times in the interval where π‘₯ 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 π‘₯ is equal to zero three times between zero and 360 degrees inclusive.

Let’s now consider the second equation, four minus one over cos π‘₯ equals zero. Multiplying through by cos of π‘₯ gives us four cos of π‘₯ minus one equals zero. We can then add one to both sides such that four cos π‘₯ equals one and, finally, divide through by four such that the cos of π‘₯ 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 π‘₯ in the interval π‘₯ is greater than or equal to zero and less than or equal to 360 degrees for which the cos of π‘₯ 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 π‘₯ 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 π‘₯ equals tan π‘₯ between zero and 360 degrees inclusive. The correct answer is five.

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