Directions: Use a graphing utility to approximate the solutions of the equation in the interval [0,2π) by setting the equation equal to zero, graphing the new equation, and using the ZERO or ROOT feature to approximate the x-intercepts of the graph. (Note: These problems are fun to graph and solve throughout the fundamental evaluation process.)
1.) 9 COS X + 2= 3
2.) 4 SIN X= COS X - 2
1.) Well, let's solve this equation and have some fun graphing it! We have 9 COS X + 2 = 3.
If we subtract 2 from both sides, we get 9 COS X = 1. Now, we need to isolate COS X, so we divide both sides by 9. Doing that gives us COS X = 1/9.
Now it's time to break out the graphing utility and have some fun! When we graph the equation COS X = 1/9, we see that it intersects the x-axis at two points in the interval [0, 2π).
Using the ZERO or ROOT feature of the graphing utility, we can approximate the x-intercepts. So, grab your graphing utility and let's clown around and find those solutions!
2.) Let's solve this equation using some humor and graphing! We have 4 SIN X = COS X - 2.
To make things more fun, let's rewrite this equation as SIN X = (COS X - 2)/4. Now, let's graph this equation using a graphing utility.
When we graph the equation SIN X = (COS X - 2)/4, we can see that it intersects the x-axis at some points in the interval [0, 2π).
Now it's time to put on our clown noses and use the ZERO or ROOT feature of the graphing utility to approximate the x-intercepts. So, grab your graphing utility and let's have some fun finding those solutions!
To approximate the solutions of the equations using a graphing utility, follow these steps:
1. For the first equation, 9 COS X + 2 = 3:
- Start by setting the equation equal to zero: 9 COS X + 2 - 3 = 0.
- Simplify the equation: 9 COS X - 1 = 0.
- Graph the new equation: y = 9 COS X - 1.
- Use the ZERO or ROOT feature on your graphing utility to approximate the x-intercepts of the graph. This will give you the approximate solutions to the equation.
2. For the second equation, 4 SIN X = COS X - 2:
- Start by rearranging the equation: 4 SIN X - COS X + 2 = 0.
- Graph the new equation: y = 4 SIN X - COS X + 2.
- Use the ZERO or ROOT feature on your graphing utility to approximate the x-intercepts of the graph. This will give you the approximate solutions to the equation.
To approximate the solutions of these equations using a graphing utility, follow these steps:
1.) Open a graphing utility software or calculator that allows plotting and solving equations.
2.) For equation 1, set 9 COS X + 2 equal to 3:
9 COS X + 2 = 3
Rearrange the equation:
9 COS X = 1
Next, subtract 2 from both sides:
9 COS X - 2 = 1 - 2
Simplify the equation:
9 COS X - 2 = -1
3.) For equation 2, set 4 SIN X equal to COS X - 2:
4 SIN X = COS X - 2
Rearrange the equation:
4 SIN X - COS X = -2
Now, you can start graphing the new equations and approximate the x-intercepts (or solutions) using the ZERO or ROOT feature of the graphing utility software or calculator:
4.) Enter the equation from step 2 into the equation field of the graphing utility and plot the graph.
5.) Look for the parts of the graph where it intersects the x-axis (y=0). Use the ZERO or ROOT feature of the graphing utility to find the x-values of these intercepts.
6.) Repeat steps 4 and 5 for the equation from step 3.
By following these steps, you can use a graphing utility to approximate the solutions of the given equations.