Question

Find the value of $\bf{2 tan^2 45\degree + sin^2 30\degree + cos^2 30\degree }$​

Answer 1

Given :-

= 2tan² 45° + Sin² 30° + Cos²30°

Here we will simply put the value of the respective Ø and we will evaluate.

= 2(1)² + (1/2)² + (√3/2)²

= 2 + 1/4 + 3/4

= (8 + 1 + 3)/4

= 12/4

= 3

Hence,

2tan² 45° + Sin² 30° + Cos²30° = 3

Some other Information :-

tan 45° = cot 45° = 1

sin 45° = cos 45° = 1/√2

tan 60° = √3

tan 30° = 1/√3

Sin 60° = √3/2

Answer 2

$\implies \sf{2 tan^2 45\degree + sin^2 30\degree + cos^2 30\degree }$

We know that :-

$\underline{ \boxed { \large \bf \: \tan(45 \degree) = 1 } } \\ \underline{ \boxed { \large \bf \: {\sin} (30 \degree) = \frac{1}{2} } } \\ \underline{ \boxed { \large \bf \: \cos(30\degree) = \frac{ \sqrt{3} }{2} } }$

$\implies \sf{2 (1)^2 + ( \dfrac{1}{2}) ^2 + (\dfrac{ \sqrt{3} }{2} )^2 }$

$\implies \sf{2 + ( \dfrac{1}{4}) + (\dfrac{ {3} }{4} ) }$

LCM = 4

$\implies \sf{\dfrac{8 + 1 + 3}{4} }$

$\implies \sf{\dfrac{12}{4} }$

$\implies \sf{\dfrac{ \cancel{12} \: \: 3}{\cancel{4} \: \: \: 1} }$

$\implies \sf3$

Answer :

$\sf{2 tan^2 45\degree + sin^2 30\degree + cos^2 30\degree } = \bf 3$

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Trigonometric table :-

$\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3} }{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }& 1 & \sqrt{3} & \rm Not \: De fined \\ \\ \rm cosec A & \rm Not \: De fined & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm Not \: De fined \\ \\ \rm cot A & \rm Not \: De fined & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}$

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Learn more:-

1. Cosθ = base / hypotenuse

2. cossecθ = 1/ sinθ

3. sec θ = 1/cosθ

4. Cotθ = 1/ tanθ

5. Sin²θ+ Cos²θ= 1

6. Sec²θ - tan²θ = 1

7. cosec ²θ - cot²θ = 1

8. sin(90°−θ) = cos θ

9. cos(90°−θ) = sin θ

10. tan(90°−θ) = cot θ

11. cot(90°−θ) = tan θ

12. sec(90°−θ) = cosec θ

13. cosec(90°−θ) = sec θ

14. Sin2θ = 2 sinθ cosθ

15. cos2θ = Cos²θ- Sin²θ