Using inverse trigonometric functions, find a solution to the equation cos(x)=0.2 in the interval 0≤x≤4π. Then, use a graph to find all other solutions to this equation in this interval. Enter your answers as a comma separated list. Thank you
x=
cosx = 0.2
x ≈ 1.37
since cosx > 0 in QI and QIV, your answers will be
x = 1.37 and 2π-1.37
To find a solution to the equation cos(x) = 0.2 using inverse trigonometric functions, we can use the arccosine function (also denoted as cos^(-1)). The arccosine function is the inverse function of the cosine function, which means it gives us the angle whose cosine is a given value.
Step 1: Find the solution using the arccosine function:
arccos(0.2) ≈ 1.3694
Step 2: Check if this solution is in the specified interval:
0 ≤ 1.3694 ≤ 4π
Since 1.3694 is in the range of 0 to 4π, it is a valid solution in the given interval.
Using a graph, we can find all the other solutions to the equation cos(x) = 0.2 in the interval 0 ≤ x ≤ 4π.
Step 1: Plot the graph of y = cos(x):
|
1.0 -|- - - - - - - - - - - - - -
|
0.8 -|- - - - -
| - - - - -
0.6 -| - - -
| - - - - -
0.4 -| - -
| - -
0.2 -| - -
|
-----------------------------
0 π 2π 3π 4π
Step 2: From the graph, identify the points where y = cos(x) intersects with y = 0.2. These intersection points represent the solutions to the equation cos(x) = 0.2.
Using the graph, it appears there are two other solutions in the interval 0 ≤ x ≤ 4π: one at approximately x = 1.3694 and the other at approximately x = 4π - 1.3694.
Therefore, the solutions to the equation cos(x) = 0.2 in the interval 0 ≤ x ≤ 4π are:
x = 1.3694, 4π - 1.3694