Consider the graph of the cosine function shown below. y=4 cos (2 x)
a. Find the period and amplitude of the cosine function.
b. At what values of θ for 0 ≤ θ ≤ 2π do the maximum value(s), minimum value(s), and zeros occur?
so i don't know how to show the graph on here but
i find the period which was π
and the amp was 4
but for the min i found x=0+ 2π * c
and the max x = π/2 + 2π * c but i cant find the zero values any tips to figure out please?
try using the unit circle as a guide
thankyou
To find the values of θ where the cosine function has zeros, we can set y = 0 in the equation y = 4 cos(2x) and solve for x.
0 = 4 cos(2x)
Dividing both sides by 4:
0 = cos(2x)
Since the cosine function has a period of 2π, we know that cos(θ) = 0 at θ = π/2 and θ = 3π/2, which correspond to x = π/4 and x = 3π/4 in the given equation.
However, because the coefficient of x in the given equation is 2, the period is compressed by a factor of 2. This means that the zeros occur at intervals of π/2, starting from x = π/4. Therefore, the values of θ where the zeros occur for 0 ≤ θ ≤ 2π are:
θ = π/4, 3π/4, 5π/4, 7π/4.
These values can be converted to x-coordinates by dividing by 2:
x = π/8, 3π/8, 5π/8, 7π/8.
To find the zero values of the cosine function, you need to solve the equation
4 cos(2x) = 0
Let's solve it step by step:
1. Divide both sides by 4:
cos(2x) = 0
2. Take the inverse cosine (also known as the arccos) of both sides:
2x = arccos(0)
3. Since the cosine of any angle whose radian measure is an integer multiple of π/2 is 0, we can write multiple solutions:
2x = π/2 + 2nπ (where n is an integer)
4. Solve for x by dividing both sides by 2:
x = (π/2)/2 + nπ/2
Simplifying:
x = π/4 + nπ/2
This gives us the general solution for the zero values of the cosine function.
For the interval 0 ≤ θ ≤ 2π, we need to find the specific values of x within this range.
Plugging in integer values for n, let's find the solutions:
When n = 0:
x = π/4
When n = 1:
x = π/4 + π/2 = 3π/4
When n = 2:
x = π/4 + (2π/2) = 5π/4
When n = 3:
x = π/4 + (3π/2) = 7π/4
So, the zero values of the cosine function within the interval 0 ≤ θ ≤ 2π are:
x = π/4, 3π/4, 5π/4, 7π/4