Question

If 5tan theta=12 than find the value of sin theta + cos theta/ cos theta +sin theta

Answer 1

QUESTION

If 5tanθ= 12 then find the value of sinθ + cosθ ÷ cosθ + sinθ ?

USED FORMULA

→» (a + b)(a - b) = a² - b²

→» b + a = a + b

ANSWER:

Firstly we have to simplify the question

Take the question

If

$5 \tan( \theta) = 12$

Then,

$\tan( \theta) = \frac{12}{5}$

method 1:

$\frac{ \sin( \theta) + \cos( \theta) }{ \cos( \theta) + \sin( \theta) }$

rationalize the denominator

$\frac{ \sin( \theta) + \cos( \theta) }{ \cos( \theta) + \sin( \theta) } \times \frac{ \cos( \theta) - \sin( \theta) }{ \cos( \theta) - \sin( \theta) }$

$\frac{ { \sin( \theta) }^{2} - { \cos( \theta) }^{2} }{ { \sin( \theta) }^{2} - { \cos( \theta) }^{2} } = 1$

without simplifying the question we will get the answer = 1

method 2:

We know that

cosθ + sinθ = sinθ + cosθ

then,

$\frac{ \sin( \theta) + \cos( \theta) }{ \cos( \theta) + \sin( \theta) } = \frac{ \sin( \theta) + \cos( \theta) }{ \sin( \theta) + \cos( \theta) }$

The numerator and denominator will get answer = 1

your answer = 1

CONCEPT USED:

→» rationalization

Answer 2

Answer:-

1

Solution:-

METHOD :- 1

Note:-

• Here I'm denoting theta with alpha
• tanA = sinA / cosA (I will be using this formula for solving)

Procedure:-

• Considering the given statement , 5tan theta = 12
• $\tan( \alpha ) = \frac{12}{5} \\ \\ \frac{ \sin( \alpha ) }{ \cos( \alpha ) } = \frac{12}{5} \\ \\ \sin( \alpha ) = 12k \: \: \: and \: \: \cos( \alpha ) = 5k$

Now considering the statement to prove,

$\frac{ \sin( \alpha ) + \cos( \alpha ) }{ \cos( \alpha ) + \sin( \alpha ) } \\ \\ = \frac{12k + 5k}{5k + 12k} \\ \\ = \frac{17k}{17k} \\ \\ = 1$

METHOD:-2

CONSIDERING GIVEN STATEMENT

$\frac{ \sin( \alpha ) + \cos( \alpha ) }{ \cos( \alpha ) + \sin( \alpha ) } \\ \\ \frac{ \sin( \alpha ) + \cos( \alpha ) }{ \sin( \alpha ) + \cos( \alpha ) } \\ \\ let \: \: \sin( \alpha ) + \cos( \alpha ) \: = x \\ \\ \frac{x}{x} \\ \\ = 1$