18.12 - Miscellaneous Examples

In the previous section, we completed a discussion on inverse trigonometric functions. In this section, we will see some miscellaneous examples.

Solved example 18.9
Find the value of   $\sin^{-1}\left(\sin \frac{3 \pi}{5}\right)$
Solution:
• We know that, sin^-1(sin x) = x. So we get $\sin^{-1}\left(\sin \frac{3 \pi}{5}\right)~=~\frac{3 \pi}{5}$
• sin^-1(sin x) is a composite function. Also, we must consider the principal branch. So this composite function is a function on $\left[- \frac{\pi}{2}, \frac{\pi}{2} \right]$. That means, both input and output must be available in the set $\left[- \frac{\pi}{2}, \frac{\pi}{2} \right]$. Details can be seen here.  
• In our present case, $\frac{3\pi}{5}$ is not available in the set $\left[- \frac{\pi}{2}, \frac{\pi}{2} \right]$.   
• So the given expression should be simplified as follows:

◼ Remarks:
1. Use identity 9(d).
   ♦ List of identities can be seen here.

Solved example 18.10
Show that $\sin^{-1}\frac{3}{5} \,-\, \sin^{-1}\frac{8}{17}\,=\,\cos^{-1}\frac{84}{85}$.
Solution:

◼ Remarks:
7.Use the identity 4.
   ♦ List of identities can be seen here.

Solved example 18.11
Show that  $\sin^{-1} \frac{12}{13} \;+\; \cos^{-1}\frac{4}{5} + \tan^{-1}\frac{63}{16} = \pi$.
Solution:
Part (i):
The given expression can be rearranged as shown below:

◼ Remarks:
3. Instead of proving the given expression, we can prove the expression in this line.

Part (ii): Simplifying the L.H.S of part(i)

 
Part (iii): Simplifying the R.H.S of part(i)

◼ Remarks:
1. Use the identity 11.
List of trigonometric identities can be seen here.
2. tan π = 0

Part (iv): Comparing L.H.S and R.H.S
• Results from parts (ii) and (iii) are the same.
• So we can write:
L.H.S of part (i) = R.H.S of part (i)

Solved example 18.12
Simplify $\tan^{-1}\left[\frac{a \cos x\,-\, b \sin x}{b \cos x\,+\, a \sin x} \right],~\text{if}~\frac{a}{b} \tan x > -1$.
Solution:

◼ Remarks:
1. We have to calculate y.
3. Divide both numerator and denominator by b cos x.
4. Use identity 11.
    ♦ List of trigonometric identities can be seen here.

Solved example 18.13
Solve $\tan^{-1} 2x \,+\, \tan^{-1} 3x \,=\,\frac{\pi}{4}$.
Solution:

◼ Remarks:
1. Use identity 10.
    ♦ List of trigonometric identities can be seen here.
6. Solve the quadratic equation for x.

◼ We obtained two values for x.
• Consider the situation where, x = -1.
• The L.H.S of the given equation will become:
tan^-1(-1) + tan^-1(-2)
• Both terms will give -ve angles.So the L.H.S will become -ve.
• But the R.H.S is a +ve angle. So x = -1 is not acceptable.
• The only solution for the given equation is: $x\,=\,\frac{1}{6}$. 

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The link below, gives some more solved examples:

Miscellaneous Exercise 

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We have completed a discussion on inverse trigonometric functions. In the next chapter, we will see matrices.

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