## Definition

Inverse Trigonometric Functions are, like any other inverse function, mathematical operators that undo another function's operation.

Given a triangle like this

the basic trigonometric functions would be defined as:

\[ \begin{array}{lcr} \sin \theta = \frac{a}{b} & \cos \theta = \frac{c}{b} & \tan \theta = \frac{a}{c} \end{array} \]

with the *angle* as their input (or argument) and a *ratio of sides* as their result. However, the inverse functions take the *ratio* as input and return the *angle*:

\[ \sin^{-1} \left( \frac{a}{b} \right) = \theta \\ \cos^{-1} \left( \frac{c}{b} \right) = \theta \\ \tan^{-1} \left( \frac{a}{c} \right) = \theta \]

This means the inverse trigonometric functions are useful whenever we know the sides of a triangle and want to find its angles.

**Note:** The notation \( \sin^{-1} \) might be confusing, as we normally use a negative exponent to indicate the reciprocal. However, in this case, \( \sin^{-1} \alpha \neq \frac{1}{\sin \alpha} \). When we want the reciprocal of \( \sin \) we use \( \csc \). In order to avoid this ambiguity, sometimes people might choose to write the inverse functions with an *arc* prefix. For example:

\[ \arccos \beta = \cos^{-1} \beta \]

## Technique

## In following statement, \( a \) and \( b \) are positive, co-prime integers. What is the sum of \( a \) and \( b \)?

\[ \cos^{-1} \frac{17}{\sqrt{1130}}= \tan^{-1} \frac{a}{b} \]

Since we are trying to find \( a \) and \( b \), we should take the tangent of both sides of the equation:

\[ \begin{align} \tan \left( \cos^{-1} \frac{17}{\sqrt{1130}} \right) &= \tan \left( \tan^{-1} \frac{a}{b} \right) \\ \tan \left( \underbrace{\cos^{-1} \frac{17}{\sqrt{1130}}}_{\large{\theta}} \right) &= \frac{a}{b} \end{align} \]

Further, we we can use the ratio given to sketch the triangle with \( \theta \) in it, using the definition of \( \cos^{-1} \):

Now, using the Pythagorean theorem, we can see that \( 17^2 + a^2 = 1130 \). This means \( a=\sqrt{1130-289}=29 \). Finally, we evaluate \( \tan \theta = \frac{29}{17} \), which means \( a+b=46 \). \( _\square \)

## Comments

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TopNewestCan you please give me inverse formulas of sine cosine tan – Anurag Suryawanshi · 3 years, 4 months ago

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