Consider the functions defined by f(x)=sin2x and g9x)=1/2tanx for x E[-90^0; 180^0]
Questions
1. Sketch the graphs of f and g on the same system of axes.
2. Calculate the x-coordinates of the points of intersection of f and g.
3. Determine the values of x for which g(x)>f(x).
To sketch the graphs of the functions f(x) = sin^2(x) and g(x) = 1/2tan(x) on the same system of axes, follow these steps:
1. Determine the range of x for which the functions are defined. In this case, it is given as x E [-90°; 180°].
2. Start by plotting the x-coordinates on the x-axis. Divide the x-axis into appropriate intervals based on the given range.
3. Calculate the corresponding y-values for each function by substituting the x-values into the respective function equations.
4. Draw the graph of f(x) = sin^2(x):
- For each x-value, calculate the value of sin^2(x).
- Plot the corresponding points on the graph.
- Connect the points with a smooth curve, matching the characteristics of a sine squared function. Remember that the graph oscillates between 0 and 1.
5. Draw the graph of g(x) = 1/2tan(x):
- For each x-value, calculate the value of 1/2tan(x).
- Plot the corresponding points on the graph.
- Connect the points with a smooth curve, considering the behavior of a tangent function. Note that for the tangent function, there are vertical asymptotes at x = -90°, 0°, and 180°.
6. Label the axes and any important points on the graph.
Next, to calculate the x-coordinates of the points of intersection of f(x) and g(x), follow these steps:
1. Identify the common values for which f(x) = g(x). These occur when the y-values of the two functions are equal.
2. Set sin^2(x) equal to 1/2tan(x) and solve for x. Use algebraic techniques to manipulate the equation and find the solutions.
3. Since these calculations can be complex, it is recommended to use a graphing calculator or a computer program to find the precise values of the intersection points. Alternatively, you can estimate the solutions by examining the graph.
To determine the values of x for which g(x) > f(x), follow these steps:
1. Compare the two functions by setting them equal to each other: g(x) = f(x).
2. Use algebraic techniques to manipulate the equation to the form: tan(x) > 2sin^2(x).
3. Solve the inequality for x. This can be done either analytically or graphically. Analytical methods involve using trigonometric identities and properties, while graphical methods would involve using a graphing calculator or a computer program.
4. Again, it is recommended to use either software or a calculator to find the exact values, or to estimate the solutions by examining the graph.