James Woods
Dennis Bowers
Math 280
10.2811
Inverse Sine and Tangent Functions
In our presentation to the class, we covered information regarding the
inverse sine and tangent functions. First, we showed that the domain and range of
the arcsin(x) respectively are -1 ≤ x ≤ 1 and –π/2 ≤ y ≤ π/2. As for the acrtan(x), the
domain and range are -∞ ≤ x ≤ ∞ and –π/2 ≤ y ≤ π/2. The domain and range are
clearly seen in the graphs below of both the arcsin(x) and arctan(x).
Next, we explained how to solve inverse sine and the inverse tangent. When
evaluating the regular sine function for right triangles, the end result is the side
opposite of the angle divided by the hypotenuse; however, when taking the inverse,
the result is an angle. When using the picture below, we know that the sine of angle
A is a/c because a is the side opposite of the angle and c is the hypotenuse of the
triangle.
When calculating the arcsin(x), one can say to themselves, “I am looking for
an angle whose sine is x.” For example the sin(π/6)=½. and the arcsin( ½ )=π/6 or
30° because you are looking for an angle whose sine is ½. Looking at the figure
above on the right, the triangle verifies the sine and inverse sine examples.
The tangent of an angle is calculated by finding the opposite side of the angle
divided by the adjacent side to the angle. Looking at the triangle above, the tangent
of angle A is a/b. When trying to find the arctan(x), the solution will be an angle just
like the arcsin(x). Taking a look at the same triangle above, the arctan(√3)=π/3 or
60°. After showing how to solve to the arcsin(x) and arctan(x) we looked at the
relationship between the original function and its inverse. There are some identities
that are very helpful when dealing with inverses. These equations are as follows:
Sin(arcsinx)=x when -1 ≤ x ≤ 1
Arcsin(sinx)=x when –π/2 ≤ x ≤ π/2
Tan(arctanx)=x when -∞ < x < ∞
Arctan(tanx)=x when –π/2 < x < π/2
Lastly, we found the derivatives of the inverse of sine and tangent. We explained
what the derivative was and we showed the differentiation of each to show where
the derivative comes from. This can be seen on the next page.
Arcsin(x)
Arctan(x)
http://en.wikipedia.org/wiki/Differentiation_of_trigonometric_functions
http://en.wikipedia.org/wiki/Inverse_trigonometric_functions