Question

if 4 tan theta =3 then prove that sin theta cos theta =12/25



plz answer me fast​

Answer 1

Solution :

$\bigstar$We have 4 tan Ф = 3

$\longrightarrow\sf{tan\theta=\dfrac{3}{4} =\bigg[\dfrac{Perpendicular}{Base} \bigg]}$

Diagram :

$\setlength{\unitlength}{1.5cm}\begin{picture}(6,2)\linethickness{0.4mm}\put(7.7,2.9){\large\sf{A}}\put(7.7,1){\large\sf{B}}\put(10.6,1){\large\sf{C}}\put(8,1){\line(1,0){2.5}}\put(8,1){\line(0,2){1.9}}\qbezier(10.5,1)(10,1.4)(8,2.9)\put(7.2,2){\large\sf{3}}\put(8.8,0.7){\large\sf{4}}\put(9.4,1.9){\large\sf{h}}\put(8.2,1){\line(0,1){0.2}}\put(8,1.2){\line(3,0){0.2}}\end{picture}$

$\underline{\boldsymbol{By\:Using\:pythagoras\:theorem\::}}}}$

$\longrightarrow\sf{(hypotenuse)^{2} =(base)^{2} +(perpendicular)^{2} }\\\\\longrightarrow\sf{(AC)^{2} =(BC)^{2} +(AB)^{2} }\\\\\longrightarrow\sf{(AC)^{2} =(4)^{2} +(3)^{2} }\\\\\longrightarrow\sf{(AC)^{2} =16+9}\\\\\longrightarrow\sf{(AC)^{2} =25}\\\\\longrightarrow\sf{AC=\sqrt{25} }\\\\\longrightarrow\sf{AC=5\:units}$

Now;

$\longrightarrow\sf{sin\theta\times cos\theta=\dfrac{12}{25} }\\\\\\\longrightarrow\sf{\dfrac{Perpendicular}{Hypotenuse} \times \dfrac{Base}{Hypotenuse} =\dfrac{12}{25} }\\\\\\\longrightarrow\sf{\dfrac{AB}{AC} \times \dfrac{BC}{AC} =\frac{12}{25}}\\\\\\\longrightarrow\sf{\dfrac{3}{5} \times \dfrac{4}{5} =\dfrac{12}{25}} \\\\\\\longrightarrow\bf{\dfrac{12}{25} =\dfrac{12}{25} \:\:\:[L.H.S=R.H.S]}$

Hence, Proved.

Answer 2

Answer:

⟶(hypotenuse)

2

=(base)

2

+(perpendicular)

2

⟶(AC)

2

=(BC)

2

+(AB)

2

⟶(AC)

2

=(4)

2

+(3)

2

⟶(AC)

2

=16+9

⟶(AC)

2

=25

⟶AC=

25

⟶AC=5units

Now;

\begin{gathered}\longrightarrow\sf{sin\theta\times cos\theta=\dfrac{12}{25} }\\\\\\\longrightarrow\sf{\dfrac{Perpendicular}{Hypotenuse} \times \dfrac{Base}{Hypotenuse} =\dfrac{12}{25} }\\\\\\\longrightarrow\sf{\dfrac{AB}{AC} \times \dfrac{BC}{AC} =\frac{12}{25}}\\\\\\\longrightarrow\sf{\dfrac{3}{5} \times \dfrac{4}{5} =\dfrac{12}{25}} \\\\\\\longrightarrow\bf{\dfrac{12}{25} =\dfrac{12}{25} \:\:\:[L.H.S=R.H.S]}\end{gathered}

⟶sinθ×cosθ=

25

12

⟶

Hypotenuse

Perpendicular

×

Hypotenuse

Base

=

25

12

⟶

AC

AB

×

AC

BC

=

25

12

⟶

5

3

×

5

4

=

25

12

⟶

25

12

=

25

12

[L.H.S=R.H.S]

Hence, Proved.