Question

Cx. (7) Write principal solutions of tan5 theta= -1
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Answer 1

Answer:

Ans:27°

Step-by-step explanation:

tan5theta= -1

tan5theta =tan(-45°)

tan5theta= tan(180°-45°)

tan5theta=tan135°

On comparing both side

5theta=135

theta=135/5

theta=27°

Answer 2

The principal solutions are

$\displaystyle \sf{ \bigg \{ \: \frac{3\pi}{20} ,\frac{7\pi}{20},\frac{11\pi}{20},\frac{15\pi}{20},\frac{19\pi}{20} , \frac{23\pi}{20} ,\frac{27\pi}{20},\frac{31\pi}{20},\frac{35\pi}{20},\frac{39\pi}{20} \: \bigg \} }$

Given : tan 5θ = - 1

To find : The principal solutions

Solution :

Step 1 of 3 :

Write down the given equation

The given equation is

tan 5θ = - 1

Step 2 of 3 :

Find the general solution set

$\displaystyle \sf{ tan \: 5 \theta = - 1}$

$\displaystyle \sf{ \implies tan \: 5 \theta = - tan \: \frac{\pi}{4} }$

$\displaystyle \sf{ \implies tan \: 5 \theta = tan \: ( \pi - \frac{\pi}{4} )}$

$\displaystyle \sf{ \implies tan \: 5 \theta = tan \: \frac{3\pi}{4} }$

$\displaystyle \sf{ \implies \: 5 \theta = k\pi + \frac{3\pi}{4} }$

$\displaystyle \sf{ \implies \: \theta = \frac{k\pi}{5} + \frac{3\pi}{20} }$

Where k is a whole number

Step 3 of 3 :

Find principal solutions

$\displaystyle \sf{ \theta = \frac{k\pi}{5} + \frac{3\pi}{20} }$

Putting k = 0 , 1 , 2 , . . . . , 9

The required principal solutions are

$\displaystyle \sf{ \bigg \{ \: \frac{3\pi}{20} ,\frac{7\pi}{20},\frac{11\pi}{20},\frac{15\pi}{20},\frac{19\pi}{20} , \frac{23\pi}{20} ,\frac{27\pi}{20},\frac{31\pi}{20},\frac{35\pi}{20},\frac{39\pi}{20} \: \bigg \} }$