On an interval of [0, 2π), can the sine and cosine values of a radian measure ever be equal? If so, enter the radian measure(s) where the values are equal. If not, enter DNE. (Enter your answers as a comma-separated list.)

Yes, they are equal in the values (in radians):

π/4, 5π/4

If cos(x) and sin(x) are defined to you as nonnegative functions (in terms of lengths), then 3π/4 and 7π/4 are also included

Step-by-step explanation:

Remember that odd multiples of 45° are special angles, with the same sine and cosine values (you can prove this, for example, by considering a right triangle with an angle of 45° and hypotenuse with length 1, and finding the trigonometric ratios).

The radian measure of 45° corresponds to π/4, hence the odd multiples on the interval [0, 2π) are π/4, 3π/4, 5π/4, 7π/4.

If you define sin(x) and cos(x) using the cartesian coordinate system (via unit circle), then cos(3π/4)=-sin(3π/4) and cos(7π/4)=-sin(7π/4). In this case, only π/4 and 5π/4 are valid choices.  

We will see that there exists a value in the given interval, such that when evaluated in it, the sine and cosine functions are equal. The value is θ = 0.785 rad

We want to see if in the interval [0, 2π) can the sine and cosine values of radian measures ever be equal.

So we basically want to solve:

sin(θ) = cos(θ)

or:

sin(θ)/cos(θ) = tan(θ) = 1

Now we can apply the inverse tangent function, Atan(x) to both sides, so we get:

Atan(tan(θ)) = Atan(1)

θ = 0.785 rad

So yes, there exists a value in the interval [0, 2π) such that the cosine and sine function are equal when evaluated in it. The value is θ = 0.785 rad

If you want to learn more, you can read:

link

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below:

80π/3 or about 83.775

step-by-step explanation:

since a cone is 1/3 of a cylinder, we need to find the base's area. since area = πr^2, we get 16π. then, multiplying this by 5, we get 80π. dividing by 3, since the cone's volume is 1/3 of that of a cylinder's, we get 80π/3.

hope this !