Solve 8cos(5x) = 4 for the three smallest positive solutions
8cos(5x) = 4
cos(5x) = 1/2
so 5x = 60° or cos 5x = 300°
x = 12 or x = 60
the period of cos 5x = 360/5 or 72°
so each of my above answers appear 72° later
from x = 12°, the next 3 answers would be 84°, 156° and 228°
from x = 60, the next 3 answers would be 132°, 204° and 276°
the three smallest answers are
12°, 60° and 84°
If you wanted the answers in radians, I assume you know how to convert.
Why did the cosine go to therapy? Because it was feeling so negative! Now, to solve the equation 8cos(5x) = 4, we can start by dividing both sides by 8, so we have cos(5x) = 4/8, which simplifies to cos(5x) = 1/2.
To find the smallest positive solutions, we'll look for the values of x where the cosine function equals 1/2.
The first solution is when the angle inside the cosine function is equal to the inverse cosine of 1/2. So, cos(5x) = 1/2 implies that 5x = cos^(-1)(1/2).
Now, I'm tempted to tell you that cos^(-1)(1/2) is a secret code for "order a pizza!" But more accurately, it means finding the angle whose cosine is 1/2. This angle is 60 degrees or π/3 radians.
So, 5x = π/3, which gives us x = π/15 as the first positive solution.
To find the next positive solutions, we can use the periodicity of the cosine function. Every 2π or 360 degrees, the cosine function repeats its values.
The second solution occurs when the angle is 2π/3 radians or 120 degrees, which gives us 5x = 2π/3. Dividing by 5, we have x = 2π/15 as the second positive solution.
Finally, the third positive solution comes when the angle is 4π/3 radians or 240 degrees. So, 5x = 4π/3, and dividing by 5, we find x = 4π/15.
Voilà! The three smallest positive solutions to 8cos(5x) = 4 are x = π/15, 2π/15, and 4π/15. Keep smiling, and remember, math can be a real trip!
To solve the equation 8cos(5x) = 4, we can divide both sides of the equation by 8:
cos(5x) = 4/8
Simplifying the right side, we have:
cos(5x) = 1/2
To find the three smallest positive solutions, we need to find the values of x that satisfy the equation in the interval [0, 2π].
Step 1: Find the first solution in the interval [0, 2π/5]:
We know that cos(π/3) = 1/2, so we have:
5x = π/3
Solving for x, we divide both sides by 5:
x = π/15
Step 2: Find the second solution in the interval [0, 2π/5]:
To find the second solution, we add the period of the cosine function to the first solution:
x = π/15 + 2π/5
Simplifying, we get:
x = π/15 + 6π/15
x = 7π/15
Step 3: Find the third solution in the interval [0, 2π/5]:
Adding the period of the cosine function again, we get:
x = π/15 + 12π/15
x = 13π/15
Therefore, the three smallest positive solutions for the equation 8cos(5x) = 4 are:
x = π/15, 7π/15, and 13π/15.
To solve the equation 8cos(5x) = 4, we need to isolate the variable x.
Step 1: Divide both sides of the equation by 8:
cos(5x) = 4/8
cos(5x) = 1/2
Step 2: Find the angle whose cosine is equal to 1/2. This can be done using the inverse cosine function (or arccosine function). The inverse cosine of 1/2 is 60 degrees.
Step 3: Since the cosine function has a repeating pattern, we can determine additional solutions by adding or subtracting a multiple of the period (360 degrees or 2π radians). The period of the cosine function is 360 degrees, divided by the absolute value of the coefficient of x, which is 5 in this case. Therefore, the period is 360/5 = 72 degrees.
Step 4: Now, let's find the three smallest positive solutions:
Solution 1: x = 60 degrees
To find the next solutions, we add the period of 72 degrees:
Solution 2: x = 60 + 72 = 132 degrees
Solution 3: x = 132 + 72 = 204 degrees
The three smallest positive solutions to the equation 8cos(5x) = 4 are x = 60 degrees, x = 132 degrees, and x = 204 degrees.