Question

tan 30 = sin 45º cos 45º + sin 30°, then find the value of 0​

Answer 1

Correct Question:

$\sf{tan \: 3\theta = sin \: {45}^{\circ} \: cos \: {45}^{\circ} + sin \: {30}^{\circ}}$

Find the value of $\bf{\theta}$

Answer :-

L.H.S = $\sf{sin \: {45}^{\circ} \: cos \: {45}^{\circ} + sin \: {30}^{\circ}}$

$\\ \\ \\ \sf{\dashrightarrow \: sin \: {45}^{\circ} \: cos \: {45}^{\circ} + sin \: {30}^{\circ}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{1}{\sqrt{2}} \times \: cos \: {45}^{\circ} + sin \: {30}^{\circ}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{1}{\sqrt{2}} \times \dfrac{1}{\sqrt{2}} + sin \: {30}^{\circ}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{1}{\sqrt{2}} \times \dfrac{1}{\sqrt{2}} + \dfrac{1}{2}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{1 \times 1}{\sqrt{2} \times \sqrt{2}} + \dfrac{1}{2}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{1}{\sqrt{4}} + \dfrac{1}{2}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{1}{\sqrt{2 \times 2}} + \dfrac{1}{2}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{1}{2} + \dfrac{1}{2}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{2}{2}}$

$\\ \\ \\ \sf{\dashrightarrow \: \dfrac{ \not{2}}{ \not{2}}}$

$\\ \\ \\ \sf{\dashrightarrow \: 1}$

We got,

$\sf{\dashrightarrow \: tan \: 3\theta = 1}$

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According the question,

• The value of $\sf{\theta}$

We know,

$\underline{\boxed{\sf{tan \: {45}^{ \circ} = 1}}}$

Therefore,

$\sf{\dashrightarrow \: 3\theta = 45}$

$\sf{\dashrightarrow \: \theta = \dfrac{45}{3}}$

$\sf{\dashrightarrow \: \theta = \cancel{\dfrac{45}{3}}}$

$\sf{\therefore \: \underline{\theta = 15}}$

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THINGS TO REMEMBER :-

• $\sf{sin \: {45}^{\circ} = \dfrac{1}{\sqrt{2}}}$

• $\sf{cos \: {45}^{\circ} = \dfrac{1}{\sqrt{2}}}$

• $\sf{sin \: {30}^{\circ} = \dfrac{1}{2}}$

• $\sf{tan \: {45}^{\circ} = 1}$

Answer 2

Step-by-step explanation:

Answer: 15º

Step-by-step explanation:

Given: tan 3x = sin 45º cos 45º + sin 30°.

To find: Value of x

Previous Knowledge:

Basic trigonometry values

0º 30º 45º 60º 90º

Sin θ 0 1/2 1/√2 √3/2 1

Cos θ 1 √3/2 1/√2 1/2 0

Tan θ 0 1/√3 1 √3 ∞

Cot θ ∞ √3 1 1/√3 0

Sec θ 1 2/√3 √2 2 ∞

Cosec θ ∞ 2 √2 2/√3 1

Solution:

➪$=> tan 3x = \frac{1}{\sqrt{2} }$

➪$×\frac{1}{\sqrt{2} } + \frac{1}{2}$

$➪=> tan 3x = \frac{1}{2}$

=> tan 3x = 1 [Since tan 45º = 1 ]

=> 3x = 45º

• $➪=> x = \frac{45}{3}$

= 15º

Thanks!