The diagram shows the curves y = 4 cosx and y = 2 + 3sin x for 0 ≤ x ≤ 2π radians. The points A and B are turning points on the curve y = 2 + 3sin x and the point C is a turning point on the curve y = 4 cosx. The curves intersect at the points D and E.
(i) Write down the coordinates of A, B and C.
(ii) Express the equation 4 cos x = 2 + 3 sin x in the form cos (x + α) = k, where α and k are constants to be found.
(iii) Hence, find in radian, the x-coordinate of D and of E.
(i) Some of you might be tempted to use differentiation to solve for the coordinates of A, B and C. While that is not wrong, it is the worst possible way to do it. It will eat up all your time for something as fundamental as this.
To find the coordinates of A and B, we must remember that the sine curve has turning points at π/2 and 3π/2. Also, the max value of 3 sinx is 3 and the min value is -3. However, since the curve has moved up the y-axis by 2 units, then the max value of 2 + 3sin x is 5 and the min value of 2 + 3sin x is -1.
Therefore, the coordinates of A and B are (π/2 , 5) and (3π/2 , -1) respectively.
For C, the min point for cos x occurs at x = π. The min value of 4 cos x = -4. Hence, the coordinates of C are (π , -4)
(ii) Given 4 cos x = 2 + 3 sin x
Therefore, 4 cos x – 3 sin x = 2
Let 4 cos x – 3 sin x ≡ R cos (x + α), where R2 = 42 + 32 = 25. Hence, R = 5
and tan α = ¾; i.e. α = 0.644 radians
Hence 4 cos x – 3 sin x ≡ 5 cos (x + 0.644).
(iii) To find the x-coordinate of D and E, therefore 5 cos (x + 0.644) = 2
cos (x + 0.644) = 0.4
x + 0.644 = 1.159 , 2π – 1.159
= 1.159 , 5.124
Therefore,
x = 1.159 - 0.644 or x = 5.124 – 0.644
= 0.515 radians = 4.45 radians
The x-coordinates of D is 0.515 rad, and the x-coordinates of E is 4.45 rad.
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