Question

if theta =30° verify the following :- (a) cos 3 theta° = 4 cos ^3 theta - 3 cos theta (B) sin 3 theta° = 3 sin theta - 4 sin ^3 theta PROPER ANSWER=BRAINLIEST

Answer 1

Solution :-

I m assume theta = x = 30°

(a) cos3x = 4cos³x - 3cosx

Putting x = 30° both sides we get,

→ cos(3*30°) = 4cos³30° - 3cosx

→ cos90° = 4cos³30° - 3cosx

values :-

• cos90° = 0
• cos30° = (√3/2)

Putting values both sides Now,

→ 0 = 4(√3/2)³ - 3(√3/2)

→ 0 = 4(3√3/8) - (3√3/2)

→ 0 = (12√3/8) - (3√3/2)

→ 0 = (3√3/2) - (3√3/2)

→ 0 = 0 (Verified) .

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(b) sin3x = 3sinx - 4sin³x

Putting x = 30° both sides we get,

→ sin(3*30°) = 3sinx - 4sin³30°

→ sin90° = 3sin30° - 4sin³30°

values :-

• sin90° = 1
• sin30° = (1/2)

Putting values both sides Now,

→ 1 = 3(1/2) - 4(1/2)³

→ 1 = (3/2) - 4(1/8)

→ 1 = (3/2) - (1/2)

→ 1 = (3 - 1)/2

→ 1 = (2/2)

→ 1 = 1 (Verified.)

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Answer 2

Solution :

We have Ф = 30°

A/q

(a) : cos 3 Ф° = 4 cos³ Ф - 3 cos Ф

Taking L.H.S :

$\mapsto\sf{cos \:3 \theta }\\\\\mapsto\sf{cos\:3(30\degree)}\\\\\mapsto\sf{cos \:90 \degree} \\\\\mapsto\sf{cos\:90\degree = 0}$

Taking R.H.S :

$\mapsto\sf{4\:cos ^{3} \theta -3\:cos \theta}\\\\\mapsto\sf{4\:(cos\:30\degree)^{3} - 3\:cos\:30\degree}\\\\\mapsto\sf{4\bigg(\dfrac{\sqrt{3} }{2} \bigg)^{3} - 3\bigg(\dfrac{\sqrt{3} }{2} \bigg)}\\\\\\\mapsto\sf{\cancel{4} \times \dfrac{3\sqrt{3} }{\cancel{8}} -\dfrac{3\sqrt{3} }{2} }\\\\\\\mapsto\sf{\dfrac{3\sqrt{3} }{2}-\dfrac{3\sqrt{3} }{2}}\\\\\\\mapsto\sf{\dfrac{3\sqrt{3} -3\sqrt{3} }{2}}\\\\\\\mapsto\sf{\dfrac{0}{2}}\\\\\mapsto\bf{0}$

∴ L.H.S = R.H.S

(b) : sin 3 Ф° = 3 sin Ф - 4 sin³ Ф

Taking L.H.S :

$\mapsto\sf{sin \:3 \theta }\\\\\mapsto\sf{sin\:3(30\degree)}\\\\\mapsto\sf{sin \:90 \degree} \\\\\mapsto\bf{sin\:90\degree = 1}$

Taking R.H.S :

$\mapsto\sf{3 \:sin\theta -4 \:sin^{3} \theta }\\\\\mapsto\sf{3\: (sin\:30\degree) - 4 \:(sin\:30\degree)^{3} }\\\\\\\mapsto\sf{3\bigg(\dfrac{1}{2} \bigg)-4\bigg(\dfrac{1}{2}\bigg)^{3} }\\\\\\\mapsto\sf{\dfrac{3}{2} -4\bigg(\dfrac{1}{8} \bigg)}\\\\\\\mapsto\sf{\dfrac{3}{2} -\cancel{\dfrac{4}{8} }}\\\\\\\mapsto\sf{\dfrac{3}{2} -\dfrac{1}{2} }\\\\\\\mapsto\sf{3-1/2}\\\\\mapsto\sf{\cancel{2/2}}\\\\\mapsto\bf{1}$

∴ L.H.S = R.H.S