Answer:
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:
https://brainly.com/question/6904750
