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Definition of Inverse Reciprocal Trig Functions

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Definition of Inverse Reciprocal Trig Functions

So far you've had to deal with trig functions, reciprocal functions, and inverse functions. Now you'll start to see inverse reciprocal functions. For example, can you compute

\sec^{-1} \frac{2}{\sqrt{3}}

As it turns out, this can be readily computed.

At the end of this Concept, you'll know how to compute this and other inverse reciprocal functions.

Watch This

Reciprocal Trig Functions

Guidance

We already know that the cosecant function is the reciprocal of the sine function. This will be used to derive the reciprocal of the inverse sine function.

y & = \sin^{-1} x\\x & = \sin y\\\frac{1}{x} & = \csc y\\\csc^{-1}  \frac{1}{x} & = y\\\csc^{-1}  \frac{1}{x} & = \sin^{-1} x

Because cosecant and secant are inverses, \sin^{-1}  \frac{1}{x}  = \csc^{-1} x is also true.

The inverse reciprocal identity for cosine and secant can be proven by using the same process as above. However, remember that these inverse functions are defined by using restricted domains and the reciprocals of these inverses must be defined with the intervals of domain and range on which the definitions are valid.

\sec^{-1}  \frac{1}{x}  = \cos^{-1} x \leftrightarrow \cos^{-1}  \frac{1}{x}  = \sec^{-1} x

Tangent and cotangent have a slightly different relationship. Recall that the graph of cotangent differs from tangent by a reflection over the y- axis and a shift of \frac{\pi}{2} . As an equation, this can be written as \cot x = -\tan \left ( x-\frac{\pi}{2} \right ) . Taking the inverse of this function will show the inverse reciprocal relationship between arccotangent and arctangent.

y & = \cot^{-1} x\\y & = -\tan^{-1} \left ( x-\frac{\pi}{2} \right )\\x & = -\tan \left( y-\frac{\pi}{2} \right )\\-x & = \tan \left ( y-\frac{\pi}{2} \right )\\\tan^{-1} (-x) & = y-\frac{\pi}{2}\\\frac{\pi}{2} + \tan^{-1} (-x) & = y\\\frac{\pi}{2} - \tan^{-1} x & = y

Remember that tangent is an odd function, so that \tan(-x) = -\tan(x) . Because tangent is odd, its inverse is also odd. So, this tells us that \cot^{-1} x = \frac{\pi}{2} - \tan^{-1} x and \tan^{-1} x =\frac{\pi}{2} - \cot^{-1} x . To graph arcsecant, arccosecant, and arccotangent in your calculator you will use these conversion identities: \sec^{-1} x = \cos^{-1}  \frac{1}{x} , \csc^{-1} x = \sin^{-1} \frac{1}{x} , \cot^{-1} x = \frac{\pi}{2} -\tan^{-1} x . Note: It is also true that \cot^{-1} x = \tan^{-1}  \frac{1}{x} .

Now, let’s apply these identities to some problems that will give us an insight into how they work.

Example A

Evaluate \sec^{-1}\sqrt{2}

Solution: Use the inverse reciprocal property. \sec^{-1} x = \cos^{-1} \frac{1}{x} \rightarrow \sec^{-1} \sqrt{2} = \cos^{-1} \frac{1}{\sqrt{2}} . Recall that \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} . So, \sec^{-1} \sqrt{2} = \cos^{-1} \frac{\sqrt{2}}{2} , and we know that \cos^{-1} \frac{\sqrt{2}}{2} = \frac{\pi}{4} . Therefore, \sec^{-1} \sqrt{2} = \frac{\pi}{4} .

Example B

Find the exact value of each expression within the restricted domain, without a calculator.

a. \sec^{-1} \sqrt{2}

b. \cot^{-1} \left ( -\sqrt{3} \right )

c. \csc^{-1} \frac{2\sqrt{3}}{3}

Solution: For each of these problems, first find the reciprocal and then determine the angle from that.

a. \sec^{-1}  \sqrt{2}  = \cos^{-1}  \frac{\sqrt{2}}{2} From the unit circle, we know that the answer is \frac{\pi}{4} .

b. \cot^{-1} \left ( -\sqrt{3} \right ) = \frac{\pi}{2} - \tan^{-1} \left ( -\sqrt{3} \right ) From the unit circle, the answer is \frac{5\pi}{6} .

c. \csc^{-1}  \frac{2\sqrt{3}}{3}  = \sin^{-1}  \frac{\sqrt{3}}{2} Within our interval, there are is one answer, \frac{\pi}{3} .

Example C

Using technology, find the value in radian measure, of each of the following:

a. \arcsin 0.6384

b. \arccos (-0.8126)

c. \arctan (-1.9249)

Solution:

a.

b.

c.

Make sure that your calculator’s MODE is RAD (radians).

Vocabulary

Inverse Function: An inverse function is a function that undoes another function.

Reciprocal Function: A reciprocal function is a function that when multiplied by the original function gives the number 1 as a result.

Guided Practice

1. \sec^{-1} (-2)

2. \cot^{-1} (-1)

3. \csc^{-1} \left ( \sqrt{2} \right )

Solutions:

1. \frac{2\pi}{3}

2. -\frac{\pi}{4}

3. \frac{\pi}{4}

Concept Problem Solution

The original goal was to evaluate \sec^{-1} \frac{2}{\sqrt{3}}

You can start with the inverse reciprocal property:

\sec^{-1} x = \cos^{-1} \frac{1}{x}

Substituting in values for "x" gives:

\sec^{-1} \frac{2}{\sqrt{3}} = \cos^{-1} \frac{1}{\frac{2}{\sqrt{3}}}

This can be rewritten as:

\cos^{-1}\frac{\sqrt{3}}{2}

And

\cos^{-1}\frac{\sqrt{3}}{2} = \frac{\pi}{6}

Therefore,

\sec^{-1} \frac{2}{\sqrt{3}} = \frac{\pi}{6}

Practice

Using technology, find the value in radian measure, of each of the following.

  1. \sin^{-1}(.345)
  2. \cos^{-1}(.87)
  3. \csc^{-1}(4)
  4. \sec^{-1}(2.32)
  5. \cot^{-1}(5.2)

Find the exact value of each expression within the restricted domain, without a calculator.

  1. \sec^{-1}(\frac{2\sqrt{3}}{3})
  2. \csc^{-1}(1)
  3. \cot^{-1}(\sqrt{3})
  4. \csc^{-1}(2)
  5. \sec^{-1}(\sqrt{2})
  6. \cot^{-1}(1)
  7. \cos^{-1}(\frac{1}{2})
  8. \sec^{-1}(2)
  9. \cot^{-1}(\frac{\sqrt{3}}{3})
  10. \sin^{-1}(\frac{\sqrt{3}}{2})

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