Question

In a right triangle ABC,AB=
$6\sqrt{3}$
BC= 6cm , & AC=12cm.
Find angle A.​

Answer 1

Answer:

⋆ DIAGRAM :

$\setlength{\unitlength}{1.5cm}\begin{picture}(6,2)\thicklines\put(7.7,2.9){\large{A}}\put(7.7,1){\large{B}}\put(10.6,1){\large{C}}\put(8,1){\line(1,0){2.5}}\put(8,1){\line(0,2){1.9}}\put(10.5,1){\line(-4,3){2.5}}\put(7,2){\large{ \sf6\sqrt{3}\:cm }}\put(9,0.7){\sf{\large{6 cm}}}\put(9.4,1.9){\sf{\large{12 cm}}}\put(8.2,1){\line(0,1){0.2}}\put(8,1.2){\line(3,0){0.2}}\end{picture}$

$\rule{150}{1}$

$\underline{\bigstar\:\textbf{According to the Question :}}$

$:\implies\sf \sin(\theta)=\dfrac{Perpendicular}{Hypotenuse}\\\\\\:\implies\sf \sin(A)=\dfrac{BC}{AC}\\\\\\:\implies\sf \sin(A) = \dfrac{6\:cm}{12\:cm}\\\\\\:\implies\sf \sin(A) = \dfrac{1}{2}\\\\\\:\implies\sf \sin(A) = \sin(30)\\\\\\:\implies\underline{\boxed{\sf A = 30^{\circ}}}$

$\rule{50}{1}\:\sf Or, Alternative\:\rule{50}{1}$

$:\implies\sf \cos(\theta)=\dfrac{Base}{Hypotenuse}\\\\\\:\implies\sf \cos(A)=\dfrac{AB}{AC}\\\\\\:\implies\sf \cos(A) = \dfrac{6\sqrt{3}\:cm}{12\:cm}\\\\\\:\implies\sf \cos(A) = \dfrac{\sqrt{3}}{2}\\\\\\:\implies\sf \cos(A) = \cos(30)\\\\\\:\implies\underline{\boxed{\sf A = 30^{\circ}}}$

⠀

$\therefore\:\underline{\textsf{Hence, Measurement of Angle A is \textbf{30 ^{\circ} }}}.$

$\rule{200}{2}$

$\bigstar\:\sf Trigonometric\:Values :\\\boxed{\begin{tabular}{c|c|c|c|c|c}Radians/Angle & 0 & 30 & 45 & 60 & 90\\\cline{1-6}Sin \theta & 0 & \dfrac{1}{2} & \dfrac{1}{\sqrt{2}} & \dfrac{\sqrt{3}}{2} & 1\\\cline{1-6}Cos \theta & 1 & \dfrac{\sqrt{3}}{2}& \dfrac{1}{\sqrt{2}}& \dfrac{1}{2}&0\\\cline{1-6}Tan \theta&0& \dfrac{1}{\sqrt{3}}&1&\sqrt{3}&Not D \hat{e} fined\end{tabular}}$

Answer 2

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$\huge\tt{GIVEN:}$

• In a right ∆ ABC,
• AB = 6√3 cm
• BC = 6 cm
• AC = 12 cm

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$\huge\tt{</strong><strong>TO~</strong><strong>F</strong><strong>I</strong><strong>N</strong><strong>D</strong><strong>:}$

• Angle A

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$</strong><strong>\</strong><strong>h</strong><strong>u</strong><strong>g</strong><strong>e</strong><strong>\tt{</strong><strong>SOLUTION</strong><strong>:}$

↪sin∅ = perpendicular / Hypotenuse

↪sin(A) = BC/AC

↪sin(A) = 6 cm/12 cm

↪sin(A) = 1/2

↪sin(A) = sin(30)

↪A = 30°

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