While evaluating the sine of a particular angle, an absent minded student failed to notice that his calculator was not in the correct angular mode. He was very lucky to end up with the right answer. What are the two smallest possible values of x such that the sine of x degrees equals the sine of x radians?

Question

While evaluating the sine of a particular angle, an absent minded student failed to notice that his calculator was not in the correct angular mode. He was very lucky to end up with the right answer. What are the two smallest possible values of x such that the sine of x degrees equals the sine of x radians?

Answer 1

0 and 180π/(180+π)

Answer 2

To determine the smallest possible values of x such that the sine of x degrees equals the sine of x radians, let's work through the problem step by step.

First, we need to understand the conversion factor between degrees and radians. There are π (pi) radians in 180 degrees, which means 1 degree is equal to π/180 radians. We can denote this as follows:

1 degree = π/180 radians

Now, let's proceed with finding the two smallest possible values of x. We can set up the equation as:

sin(x degrees) = sin(x radians)

Using the conversion factor, we can express x degrees in terms of radians as well. Recall that 1 degree is equal to π/180 radians. Therefore, x degrees is equal to x * (π/180) radians.

Substituting this into our equation, we have:

sin(x * (π/180)) = sin(x)

Now, we can use a trigonometric identity to simplify the equation further. The identity states that sin(a) = sin(π - a). In our equation, this means that sin(x) = sin(π - x * (π/180)). Therefore, we can rewrite the equation as:

sin(x * (π/180)) = sin(π - x * (π/180))

To find the two smallest possible values of x, we need to solve this equation.

The first possible value is when:

x * (π/180) = π - x * (π/180)

We can solve this equation by bringing all the x terms to one side:

2 * x * (π/180) = π

Multiplying both sides by 180/π, we get:

2 * x = 180

Dividing both sides by 2, we find:

x = 90 degrees

Now, let's find the second possible value by considering the case when:

x * (π/180) = x * (π/180) - π

Again, by rearranging the equation, we have:

0 = x * (π/180) - x * (π/180) + π

Simplifying, we get:

0 = π

This equation is not solvable, as π is a constant value and cannot be canceled out. Therefore, there is no second possible value for x in this case.

In conclusion, the two smallest possible values for x such that the sine of x degrees equals the sine of x radians are:

x = 90 degrees