Question

Today's Question :-
$\looparrowright {\sin}^{2}30° \: { \cos}^{2}45° + 4 {\tan}^{2}30° + \frac{1}{2} { \sin}^{2}90°$
❍ Solve it no direct answers.
❍ For Moderator, Maths Aryabhatta and Best Users.
❍ All the best!!!
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Answer 1

⇝ Equation :-

$\small{{\sin}^{2}30° \: { \cos}^{2}45° + 4 {\tan}^{2}30° + \frac{1}{2} { \sin}^{2}90°}$

❍ Solution :-

We know that,

⇝ sin 30° = $\frac{1}{2}$

⇝ cos 45° = $\frac{1}{\sqrt{2}}$

⇝ tan 30° = $\frac{1}{\sqrt{3}}$

⇝ sin 90° = 1

Now, putting values in the equation, we get,

$\sf→ {(\frac{1}{2})}^{2} \times{(\frac{1}{\sqrt{2}})}^{2} + 4{(\frac{1}{\sqrt{3}})}^{2} + \frac{1}{2}{(1)}^{2}$

$\sf→ \frac{1}{4} \times \frac{1}{2} + \frac{4}{3} + \frac{1}{2}$

$\sf→ \frac{1}{8} + \frac{4}{3} + \frac{1}{2}$

⇝ LCM (8, 3, and 2)= 24

$\sf→ \frac{(1 \times 3) + (4 \times 8) + (1 \times 12)}{24}$

$\sf→ \frac{3 + 32 + 12}{24}$

$\sf→ \frac{47}{24}$

Hence, sin²30° cos²45° + 4tan²30° + $\frac{1}{2}$sin²90° is equal to 47/24.

Answer 2

Solution :

sin² 30 cos² 45 + 4 tan² 30 + ½ sin² 90

Let sin 30 = a , cos 30 = b and cos 45 = c .

sin 90 is 1

>> a² cos 45 + 4(a²/b²) + ½

>> (a²/√2) + 4(a²/b²) + ½

sin 30 = ½ = a

&

cos 30 = √3/2 = b

a²/b²

>> (1/4) / (3/4)

>> ⅓

>> (1/4√2) + 4×⅓ + ½

>> 1/(4√2) + (4/3) + ½

>> [3 + 16√2 + 6√2 ] / (12√2)

>> [ 22√2+3]/(12√2)

>> (22/12) + 1/(4√2)

>> 11/6 + (√2/8)

>> 1 ⅚ + ⅛√2 .

This is the required answer .

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Additional Information :

sin ²theta + cos ²theta = 1

tan²theta + 1 = sec² theta

cot²theta + 1 = cosec² theta

$\large{ \begin{tabular}{|c|c|c|c|c|c|} \cline{1-6} \theta & \sf 0^{\circ} & \sf 30^{\circ} & \sf 45^{\circ} & \sf 60^{\circ} & \sf 90^{\circ} \\ \cline{1-6} \sin & 0 & \dfrac{1}{2 } & \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} & 1 \\ \cline{1-6} \cos & 1 & \dfrac{ \sqrt{ 3 }}{2} } & \dfrac{1}{ \sqrt{2} } & \dfrac{ 1 }{ 2 } & 0 \\ \cline{1-6} \tan & 0 & \dfrac{1}{ \sqrt{3} } & 1 & \sqrt{3} & \infty \\ \cline{1-6} \cot & \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } &0 \\ \cline{1 - 6} \sec & 1 & \dfrac{2}{ \sqrt{3}} & \sqrt{2} & 2 & \infty \\ \cline{1-6} \csc & \infty & 2 & \sqrt{2 } & \dfrac{ 2 }{ \sqrt{ 3 } } & 1 \\ \cline{1 - 6}\end{tabular}}$

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