Question

find the value of sin^2 45°+ cos^2 45°÷ tan^2 60°​

Answer 1

We have to find out the value of :-

$\sf \implies{sin}^{2} 45 \degree + {cos}^{2} 45\degree \div {tan}^{2} 60\degree$

We know that :-

$\bf \large \: sin \: 45\degree = \dfrac{1}{ \sqrt{2} }$

$\bf \large \: cos \: 45\degree = \dfrac{1}{ \sqrt{2} }$

$\bf \large \: tan \: 60\degree = \sqrt{3}$

Now put these values.

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

$\sf \implies{ \dfrac{1}{{2} } } + \dfrac{1}{{2} } \div {3}$

Now using BODMAS :-

$\sf \implies{ \dfrac{1}{{2} } } + \dfrac{1}{{2} } \times \dfrac{1}{3}$

$\sf \implies{ \dfrac{1}{{2} } } + \dfrac{1}{{6} }$

LCM of 2 and 6 = 6

$\sf \implies \dfrac{3 + 1}{{6} }$

$\sf \implies \dfrac{4}{{6} } = \dfrac{2}{{3} }$

Answer :

$\sf {sin}^{2} 45 \degree + {cos}^{2} 45\degree \div {tan}^{2} 60\degree = \bf \large \dfrac{2}{3}$

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Trigonometric table :-

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

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

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About BODMAS :-

BODMAS is used to simplify the expressions having different operators (+,-,÷or ×) . It is basically a specific order.

B => Brackets

O => Orders or Of

D => Division

M => Multiplication

A => Addition

S => Subtraction

This means start solving first anything which is inside bracket. Then evaluate any root or power. Then do division , multiplication , addition and subtraction in the sequence of BODMAS.