Question

in triangle ABC Ab=6√3cm,Ac=12cm,bc=6cm. find measure of angle A​

Answer 1

Question:

In triangle ABC AB=$\mathsf{6\sqrt{3}}cm$, AC=12cm, BC=6cm. find measure of angle A.

Given:

AB=$\mathsf{6\sqrt{3} cm}$

AC=12 cm

BC=6 cm

To Find:

Measure of $\angle{A}$

Solution:

Here,

$\dashrightarrow\mathsf{{AB}^{2}+{BC}^{2}={(6\sqrt{3)}}^{2}+{(6)}^{2}}$ $\: \: —(1)$

We know,

$\mathsf{{AC}^{2}={12}^{2}=144\: {cm}^{2}} \: \: —(2)$

From (1) & (2),

$\mathsf{AB}^{2}+{BC}^{2}={AC}^{2}$

We also know, that AB²+BC²=AC² is Pythagoras theorem that implies on right angled Triangle, that means ∆ABC is right angled Triangle. So,

$\implies\angle{B}$

Now,

$\dashrightarrow\mathsf{tanA=\large\frac{BC}{AB}}$

$\dashrightarrow\mathsf{tanA=\large\frac{6}{6\sqrt{3}}}$

$\dashrightarrow\mathsf{tanA=\large\frac{1}{\sqrt{3}}cm}$

We know that, ${tan30}{o}=\frac{1}{\sqrt{3}}$. So,

$\mathsf{\angle{A}={30}^{o}}$

Hence, the measure of $\angle{A}$ is 30°

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More to Know:

Trignometric Ratios -:

$>\mathsf{sin\theta=\large\frac{perpendicular}{Hypotenuse}}$

$>\mathsf{cos\theta=\large\frac{Base}{Hypotenuse}}$

$>\mathsf{tan\theta=\large\frac{Perpendicular}{Base}}$

$>\mathsf{cosec\theta=\large\frac{Hypotenuse}{Perpendicular}}$

$>\mathsf{sec\theta=\large\frac{Hypotenuse}{Base}}$

$>\mathsf{cot\theta=\large\frac{Base}{Perpendicular}}$

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Answer 2

GivEn:

• AB = 6√3cm
• AC = 12 cm
• BC = 6 cm

To find:

• The measure of angle A?

Solution:

• Let's consider ABC is a triangle.

Where,

• Hypotenuse = 12 m = AC
• Base = 6√3 cm = AB
• Perpendicular = 6 cm = BC

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« Now, By using Pythagoras Theorem,

→ (Hypotenuse)² = (Perpendicular)² + (Base)²

→ (12)² = (6)² + (6√3)²

→ 144 = 36 + 108

→ 144 = 144

∴ Hence, It is a right angled triangle.

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« Now, Let's find angle A,

→ Tan A = per/base

→ Tan A = 6/6√3

By cancelling 6,

→ Tan A = 1/√3

→ Tan A = 30°

∴ Hence, The measure of angle A = 30°.

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★ More to know:

Trigonometric Identities:

• sin²θ + cos²θ = 1
• sec²θ - tan²θ = 1
• csc²θ - cot²θ = 1

Trigonometric relations:

• sinθ = 1/cscθ
• cosθ = 1 /secθ
• tanθ = 1/cotθ
• tanθ = sinθ/cosθ
• cotθ = cosθ/sinθ

Trigonometric ratios:

• sinθ = opp/hyp
• cosθ = adj/hyp
• tanθ = opp/adj
• cotθ = adj/opp
• cscθ = hyp/opp
• secθ = hyp/adj