Let's analyze and solve the given problems based on the graph provided and the function $f(x) = 2 \cos 2x - 0.5$.

Problem 1(a)

What are the values of the amplitude and period of the function $f(x) = 2 \cos 2x - 0.5$?

Amplitude: The amplitude of a cosine function $a \cos(bx + c) + d$ is given by the absolute value of $a$. For the given function: $f(x) = 2 \cos 2x - 0.5$ The coefficient of the cosine function is 2, so the amplitude is: $\text{Amplitude} = |2| = 2 \text{ units}$

Period: The period of a cosine function $a \cos(bx + c) + d$ is given by: $\text{Period} = \frac{2\pi}{|b|}$ For the given function, $b = 2$, so the period is: $\text{Period} = \frac{2\pi}{2} = \pi \text{ units}$

Problem 1(b)

Find the $x$-coordinates of the points of intersection for $0^\circ \leq x \leq 360^\circ$ between $f(x) = 2 \cos 2x - 0.5$ and $g(x) = \sin x$.

To find the points of intersection, set $f(x) = g(x)$: $2 \cos 2x - 0.5 = \sin x$

This equation can be solved numerically or graphically as it involves both $\cos 2x$ and $\sin x$.

1. Use the double-angle identity for cosine: $\cos 2x = 2 \cos^2 x - 1$ Thus, the equation becomes: $2 (2 \cos^2 x - 1) - 0.5 = \sin x$ $4 \cos^2 x - 2 - 0.5 = \sin x$ $4 \cos^2 x - 2.5 = \sin x$

2. Solve this numerically by finding the values of $x$ where this equation holds in the interval $0^\circ \leq x \leq 360^\circ$.

Since this is a complex equation to solve algebraically, we will consider it graphically or use numerical methods.

From the graph: Let's approximate the $x$-coordinates of the intersections. Observing the graph of $2 \cos 2x - 0.5$ and comparing it with $\sin x$, we can estimate the intersections:

• At approximately $x \approx 38.0^\circ$
• At approximately $x \approx 142.0^\circ$
• At approximately $x \approx 218.0^\circ$
• At approximately $x \approx 322.0^\circ$

Answers: $x = 38.0^\circ, 142.0^\circ, 218.0^\circ, 322.0^\circ$

Summary

1. Amplitude: 2 units
2. Period: $\pi$ units
3. Intersections: $x = 38.0^\circ, 142.0^\circ, 218.0^\circ, 322.0^\circ$

If you have any questions or need further details, feel free to ask!

Related Questions:

1. What is the general formula for the amplitude of a trigonometric function?
2. How do you determine the period of a trigonometric function?
3. What are the double-angle identities in trigonometry?
4. How do you solve trigonometric equations graphically?
5. How can you find the intersection points of two trigonometric functions numerically?
6. What is the significance of the phase shift in trigonometric functions?
7. How do trigonometric identities help in solving complex trigonometric equations?
8. What methods can be used to approximate solutions of trigonometric equations?

Tip:

To solve complex trigonometric equations involving multiple angles and different functions, consider using graphical or numerical methods when algebraic manipulation becomes too cumbersome.