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

10cm

step-by-step explanation:

ik