Inverse trigonometric functions are the opposite of regular trigonometric functions.

Regular functions like sine, cosine, and tangent help us find ratios in a right triangle based on the angles.

On the other hand, inverse functions help us find the angles when we have those ratios!

This is really important in different fields, like geometry, physics, and engineering.

Definitions:

• Sine Inverse: This is written as $\sin^{-1}(x)$ or $\arcsin(x)$. It gives us the angle that has a sine of $x$.

• Cosine Inverse: Notated as $\cos^{-1}(x)$ or $\arccos(x)$. This tells us the angle with a cosine of $x$.

• Tangent Inverse: Denoted as $\tan^{-1}(x)$ or $\arctan(x)$. It finds the angle that has a tangent of $x$.

Applications:

Let's say you have a right triangle. You know the opposite side is 3 units long and the hypotenuse is 5 units long.

You can use the sine function to find the ratio:

$$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}$$

Then, to find the angle $\theta$, you use the sine inverse:

$$\theta = \sin^{-1}\left(\frac{3}{5}\right)$$

Being able to find angles from known ratios is really important in areas like architecture, navigation, and even robotics!