QuestionFind the exact value of $\sin 1935^{\circ}$ . Simplify your answer using integers or fractions.

Studdy Solution

STEP 1

Assumptions1. The angle is given in degrees. . We are asked to find the exact value of the sine function for the given angle.
3. We know that the sine function is periodic with a period of $360^{\circ}$ .

STEP 2

First, we need to reduce the given angle to an equivalent angle within one period of the sine function. We can do this by using the property of periodicity of the sine function.
$\sin x = \sin (x +360n)$ where $n$ is an integer.

STEP 3

Now, we need to find the equivalent angle within one period of the sine function for the given angle. We can do this by dividing the given angle by $360^{\circ}$ and taking the remainder.
$Equivalent\, angle =1935^{\circ} \mod360^{\circ}$

STEP 4

Calculate the equivalent angle.
$Equivalent\, angle =193^{\circ} \mod360^{\circ} =155^{\circ}$

STEP 5

Now that we have the equivalent angle within one period of the sine function, we can find the exact value of the sine function for this angle. We know that $\sin155^{\circ}$ is in the second quadrant where sine is positive and equals to $\sin (180^{\circ} -155^{\circ})$ .
$\sin155^{\circ} = \sin (180^{\circ} -155^{\circ})$

STEP 6

Calculate the angle inside the sine function.
$\sin155^{\circ} = \sin (180^{\circ} -155^{\circ}) = \sin25^{\circ}$

STEP 7

Now, we need to find the exact value of $\sin25^{\circ}$ . Unfortunately, $\sin25^{\circ}$ is not a standard angle and its exact value is not a simple number or a simple radical. Therefore, we cannot simplify it further.
So, the exact value of $\sin1935^{\circ}$ is $\sin25^{\circ}$ .

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