Evaluate: sin^-1(sin 100) + cos^-1(cos 100) + tan^-1(tan 100) + cot^-1(cot 100).

To evaluate the expression sin^-1(sin 100) + cos^-1(cos 100) + tan^-1(tan 100) + cot^-1(cot 100), we need to consider the properties of the inverse trigonometric functions.

Understanding Each Component

• sin^-1(sin 100): The output of this function is the angle in the range of -90° to 90°. Since 100° is outside this range, we find the equivalent angle: 100° - 180° = -80°.
• cos^-1(cos 100): This function returns an angle between 0° and 180°. The equivalent angle for 100° is 100° itself.
• tan^-1(tan 100): The tangent function is periodic with a period of 180°. Thus, tan^-1(tan 100) = 100° - 180° = -80°.
• cot^-1(cot 100): Similar to tangent, this function also has a periodicity. Therefore, cot^-1(cot 100) = 100°.

Combining the Results

Now, we can sum the evaluated components:

sin^-1(sin 100) = -80°

cos^-1(cos 100) = 100°

tan^-1(tan 100) = -80°

cot^-1(cot 100) = 100°

Adding these together:

-80° + 100° - 80° + 100° = 40°

The Final Answer

The value of the expression is 40°.