Question

trigonometric problem
​

Answer 1

Question:-

$\rm \:if \: \tan(A) = \frac{3}{4} \: then \: \sin(A) \cos(A) = \frac{12}{25}$

Solution:-

$\rm \: \tan(A) = \frac{3}{4} = \frac{p}{b}$

$\rm \: p = 3 \: \: \: b =4 \: \: and \: h = x$

Using phythogoeros theorem

$\rm \: {h}^{2} = {p}^{2} + {b}^{2}$

$\rm \: {x}^{2} = {3}^{2} + {4}^{2}$

$\rm \: {x}^{2} = 9 + 16$

$\rm \: {x}^{2} = 25$

$\rm \: x = 5$

So

$\rm \: p = 3 \: \: \: b =4 \: \: and \: h = 5$

$\rm \: \sin(A) = \frac{p}{h} = \frac{3}{5}$

$\rm \: \cos(A) = \frac{b}{h} = \frac{4}{5}$

So we have

$\rm \: \sin(A) \cos(A) = \frac{12}{25}$

$\rm \: \frac{3}{5} \times \frac{4}{5} = \frac{12}{25}$

$\rm \: \frac{12}{25} = \frac{12}{25}$

Hence proved

Answer 2

Answer:

Question:-

\rm \:if \: \tan(A) = \frac{3}{4} \: then \: \sin(A) \cos(A) = \frac{12}{25}iftan(A)=

4

3

thensin(A)cos(A)=

25

12

Solution:-

\rm \: \tan(A) = \frac{3}{4} = \frac{p}{b}tan(A)=

4

3

=

b

p

\rm \: p = 3 \: \: \: b =4 \: \: and \: h = xp=3b=4andh=x

Using phythogoeros theorem

\rm \: {h}^{2} = {p}^{2} + {b}^{2}h

2

=p

2

+b

2

\rm \: {x}^{2} = {3}^{2} + {4}^{2}x

2

=3

2

+4

2

\rm \: {x}^{2} = 9 + 16x

2

=9+16

\rm \: {x}^{2} = 25x

2

=25

\rm \: x = 5x=5

So

\rm \: p = 3 \: \: \: b =4 \: \: and \: h = 5p=3b=4andh=5

\rm \: \sin(A) = \frac{p}{h} = \frac{3}{5}sin(A)=

h

p

=

5

3

\rm \: \cos(A) = \frac{b}{h} = \frac{4}{5}cos(A)=

h

b

=

5

4

So we have

\rm \: \sin(A) \cos(A) = \frac{12}{25}sin(A)cos(A)=

25

12

\rm \: \frac{3}{5} \times \frac{4}{5} = \frac{12}{25}

5

3

×

5

4

=

25

12

\rm \: \frac{12}{25} = \frac{12}{25}

25

12

=

25

12

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