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Using Rcos(x + α) to Find the Maximum and Minimum Values of a Function and to Solve a Trigonometric Equation

The mathematical solution initially explains how to find the maximum and minimum values of f(x) = 5sin x + 2cos x and how to find, in radians to 2 decimal places, the smallest positive values of x at which they occur. f(x) is written in the form Rcos(x + α). Explanation is then given as to how the maximum and minimum values are found from an expression written in this form together with the values of x at which these points occur. The second part of the explanation shows how the Rcos(x + α) form can be used to solve the equation 5sin x + 2cos x = 1. The solution uses the fact that cos(x) = cos(-x) to find the solutions in the required range.

The graphical solution 1 explains how to use the graphic calculator to find the coordinates of the maximum and minimum values of the given function. The function is drawn and the coordinates of the maximum and minimum points are verified.

The graphical solution 2 explains how to use graphic calculator to solve the equation 5sin x + 2cos x = 1 by drawing the graph of y = 5sin x + 2cos x and the graph of y = 1 and finding the points of intersection. The explanation explains how to set the axes to ensure the solutions are shown on the screen.

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