Question

if sin A=3/4,find the value of 2+2 tan^2 A.
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Answer 1

Step-by-step explanation:

given sinA= 3/4

then CosA =root7/4

cos^2=7/16

sec^2A=16/7

2(1+tan^2A)

=2sec^2A

=2*16/7

=32/7

Answer 2

Given :

sinA = (3/4)

To find :

Value of , 2 + 2 tan²A

Identity used :

$\sf \bullet \: \: \cos(x) = \sqrt{1 - { \sin(x) }^{2} } \\ \\ \sf \bullet \: \: \tan(x) \: = \: \dfrac{ \sin(x) }{ \cos(x) }$

Solution :

First of all we need to find cosA

$\sf \implies \: \cos(A) \: = \: \sqrt{ \: 1 - { \sin(A) }^{2} } \\ \\ \sf \implies \: \cos(A) \: = \sqrt{ \: 1 \: - \: { \bigg \{ \dfrac{3}{4} \bigg \} }^{2} } \\ \\ \sf \implies \: \cos(A) \: = \: \sqrt{ \: 1 - \dfrac{9}{16} } \\ \\ \sf \implies \: \cos(A) \: = \: \sqrt{ \dfrac{(1 \times 16) - (9 \times 1)}{16} } \\ \\ \sf \implies \: \cos(A) \: = \: \sqrt{ \dfrac{16 - 9}{16} } \\ \\ \sf \implies \: \cos(A) \: = \: \sqrt{ \dfrac{7}{16} } \\ \\ \sf \implies \: \cos(A) \: = \: \dfrac{ \sqrt{7} }{4}$

Now we need to find tan A

$\sf \implies \: \: \tan(A) \: = \: \dfrac{ \sin(A) }{ \cos( A ) } \\ \\\sf \implies \: \: \tan(A) \: = \: \dfrac{ \dfrac{3}{4} }{ \dfrac{ \sqrt{7} }{4} } \\ \\ \sf \implies \: \: \tan(A) \: = \: \dfrac{3}{4} \times \dfrac{4}{ \sqrt{7} } \\ \\ \sf \implies \: \: \tan(A) \: = \: \dfrac{3}{ \sqrt{7} }$

Now value of 2 + 2 tan²A =

$\sf \implies \: 2 \: + \: 2 \times { \bigg \{ \dfrac{3}{ \sqrt{7} } \bigg \} }^{2} \\ \\ \sf \implies \: 2 \: + \: 2 \times \dfrac{9}{7} \\ \\ \sf \implies \: 2 \: + \: \dfrac{18}{7} \\ \\ \sf \implies \: \dfrac{(2 \times \: 7 ) + (18 \times 1)}{7} \\ \\ \sf \implies \: \dfrac{14 + 18}{7} \\ \\ \sf \implies \: \dfrac{32}{7}$

ANSWER :

2 + 2 tan²A = 32/7