WMA11 Jan 2019

5. WMA11/01 Edexcel IAL P1 new January 2019, Q5 (Trignometric Ratios: Transforming Trignometric Functions)

Figure 2 shows a plot of part of the curve with equation y = cos 2x with x​​ being measured in radians.​​ 

The point P, shown on Figure 2, is a minimum point on the curve.​​ 

(a) State the coordinates of P.​​ 

(2)

A copy of Figure 2, called Diagram 1, is shown at the top of the next page.​​ 

(b) Sketch, on Diagram 1, the curve with equation y = sin x​​ 

(2)

Hence, or otherwise, deduce the number of solutions of the equation

(i)​​ cos⁡2x = sin⁡x​​ that lie in the region​​ 0 ≤ x≤ 20π​​ 

​​ (ii)​​ cos⁡2x = sin⁡x​​ that lie in the region​​ 0 ≤ x≤ 21π​​ 

(2)

SOLUTION​​ 

a-​​ Since the given graph is a stretch graph of f(x), the x-cordinates are multiplied by the scale factor of ½.​​ 

Thus, the coordinates of P are​​ 

Pπ2, -1

b- i-​​ Let’s first sketch the sine graph on the same grid.​​ 

Now, mark the points of intersection of cosine and sine graphs.​​ 

Hence, there are about 3 solutions in a period of​​ 0 to 2π. Thus, in a region of​​ 0≤x≤20π, there will be about total 10 periods of​​ 2π. Hence, the total number of solutions​​ that lie in the region​​ 0≤ x≤20π​​ are

3×10=30 solutions

b- ii-​​ We have already found that the number of solutions between ​​ 0≤x≤20π​​ are 30. Now, for the region​​ 0≤x≤21π, the number of solution would be 30 plus the number of solution between 20π to 21π. Remember, that the trignometric graphs are symmetric therefore the number of solution between 20π to 21π would be the same as the number of soliutions between 0π to π. And, from the sketch​​ it can be read that there are 2 solutions between 0π to π which would be same for the region between 20π to 21π.​​ 

Hence, the total number of solutions between 0π to 21π are

=30+2

=32 solutions