Evaluate the trigonometric function using its period as an aid.
cos 5pi
how would i go about finding the period and what is a period?
Period: When does it repeat itself?
Cosine repeats each 2PI, so
cos5Pi= cos(5PI-2PI)=cos3PI= cos(PI+2PI)= cos PI.
Cos 5 pi
The period of a trigonometric function is the length of one complete cycle of the function. For cosine and sine functions, the period is typically 2π.
To find the period of a trigonometric function, you need to observe how the function repeats itself over a certain range of values.
In the case of cosine (cos) function, the general form is:
cos(x)
The cosine function repeats itself every 2π radians. This means that the cosine of any angle x is equal to the cosine of that angle plus any multiple of 2π.
Now, to evaluate the trigonometric function cos(5π), you need to determine the equivalent angle within one period.
Step 1: Identify the given angle
The given angle is 5π.
Step 2: Determine the equivalent angle within one period
Since one period of cos(x) is 2π, you can find the equivalent angle within one period by dividing the given angle by 2π:
Equivalent angle = 5π ÷ 2π = 5/2
Step 3: Calculate the function value
Now that you have the equivalent angle within one period, you can evaluate the trigonometric function cos(5π) as cos(5/2) within that range.
So, the final step is to calculate the value of cos(5/2).
To find the period of a trigonometric function, you need to understand what the term "period" means. In trigonometry, the period of a function is the smallest positive value for which the function repeats itself.
For the cosine function (cos), the period is always equal to 2π (or 360 degrees). This means that cos(x) = cos(x + 2π), for any value of x. Knowing this, we can use the period to evaluate the trigonometric function.
In this case, you are asked to evaluate cos(5π). To do this, we can utilize the fact that the period is 2π. Since 5π is greater than one period, we need to find an equivalent angle within one period.
To do this, we subtract 2π from 5π until we obtain an angle within the first period. Here's the calculation:
5π - 2π = 3π
Hence, evaluating cos(5π) is equivalent to evaluating cos(3π), as both represent the same angle within the first period.
Now, we can evaluate the trigonometric function cos(3π). The cosine function evaluates to -1 at 180 degrees or π radians. Since 3π is equal to 3 times π, which is equal to 180 degrees, we have:
cos(3π) = cos(180 degrees) = -1
So, the value of cos(5π) is -1.