A common problem in trigonometry is finding an angle whose trig functions are known.

Problems of this kind involve the computation of **bow functions**such as arcsen *x*arccos* x*arctg *x*, and so on. Consider this idea from the point of view of inverse functions, with the goal of developing derivative formulas for inverse trigonometric functions.

## Identities for inverse trigonometric functions

If we interpret *x* as an angle measured in radians whose sine is *x*, and if that angle is *not negative, *then we can represent *x* as an angle in a right triangle, where the hypotenuse has length 1 and the opposite side to the angle of has length *x *(** figure a**). By the Pythagorean Theorem, the adjacent side to the angle has length .

In addition, the angle opposite to é , since the cosine of that angle is *x *(* figure b*). This triangle motivates several useful identities, involving trigonometric functions that are valid for . For example:

Similarly, *x* and *x* can be represented with angles of right triangles shown in ** figure c and d.** These triangles reveal more useful identities, such as:

**NOTE. **Nothing is gained by memorizing these identities; what is important is to understand the * method* used to get them.

**Example**

* The figure* below shows a computer generated graph of

*y =*(sen

*x*). You might think this graph should be the straight line

*y = x*, once (sen

*x)*=

*x*. Why does not this happen?

** Solution. **The relationship (sen

*x*) =

*x*is valid in range ; soon we can say for sure that the graphs of

*y = (sen*

*x*) and*y = x*coincide in this range. However, outside this range, the relationship (sen*x*) =*x*It does not have to be valid. For example, if you are in the range then the amount*x*- will be in range . Like this*
*

*
*

Thus, using the identity sen (*x*-) = -sen * x* and the fact that it's an odd function we can express (sen *x*) how

This shows that in the range , the graph of * y = (sen x) coincides with the line y = -(x-), which has slope -1 and an intercept x in x = , which is in accordance with The figure.*

*Derivation formula*

*Remember that if f is a one-to-one function whose derivative is known, so there are two basic ways to get a derivation formula for (x), we can rewrite the equation y = (x) how x = f(y), and differentiate implicitly. We will use implicit differentiation to obtain the derivation formula for y = x. Rewriting this equation as x = sen y and differentiating implicitly we get *

*This derivative formula can be simplified by applying the formula , which was deduced from the triangle of the figure, resulting:*

*Thus, we show that *

*If u is a differentiable function of x, then and the chain rule produce the following generalized derivative formula*

*The method used to obtain this formula can also be used to obtain generalized derivative formulas for other inverse trig functions. These formulas, valid for -1 < u <1, are*

* *

*Next content: Series and Sequences*