find each value by referring to the graphs of the trig functions.
1. sec 4 pi...i got 0
Find the values for theta for which each equation is true.
csc (theta)=1 i got pi over 2 +2 pi n
To find the value of sec(4π), we can start by understanding the graph of the secant function. The secant function is the reciprocal of the cosine function, so wherever the cosine function has a value of zero, the secant function will be undefined.
In the unit circle, the cosine function is zero at π/2 and 3π/2. Since the secant function is undefined at these points, we need to look for the next available point where sec(θ) is defined.
To simplify, we can convert 4π into an angle within the range of 0 to 2π by dividing 4π by 2π, which gives us 2. So, 4π is equivalent to 2 times around the unit circle.
Since the secant function has a period of 2π, we can find an equivalent angle within the range of 0 to 2π by subtracting 2π from 4π, which gives us 2π. Therefore, sec(4π) is equivalent to sec(2π).
From the unit circle, we know that cos(2π) = 1. And since sec(θ) is the reciprocal of cos(θ), sec(2π) will also be 1.
Hence, sec(4π) = sec(2π) = 1.
Moving on to the second question, to find the values of theta where csc(theta) is equal to 1, we can analyze the graph of the cosecant function.
The cosecant function is the reciprocal of the sine function, and it is equal to 1 when the sine function has a value of 1. The sine function takes the value of 1 at π/2 and 5π/2.
To find additional values of theta where csc(theta) = 1, we can use the periodicity of the cosecant function. The cosecant function repeats itself every 2π.
So, the initial values we found (π/2 and 5π/2) can be used to generate additional solutions by adding or subtracting multiples of 2π. In this case, it would be π/2 + 2πn, where n is an integer.
Putting it all together, the values of theta for which csc(theta) is equal to 1 are:
θ = π/2 + 2πn, where n is an integer.
Note: It is important to note that when using trigonometric functions, the angles are often expressed in radians.