Question

1 sf sin o = √3/2
find the value of all T- ratios

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Answer 1

Correct Question :-

If $\sin \theta = \frac{ \sqrt{3} }{2}$ then, find the value of all T - ratios.

Step by step explanation :-

First see the attached diagram to the answer.

Let us draw the right angled triangle ∆ABC, ∠B = 90° and ∠A = θ.

We know that,

$\rm{ \sin \theta = \frac{perpendicular}{hypotenuse} }$

Hence,

$\rm{ \sin \theta = \frac{ \sqrt{3} }{2} }$

By Pythagoras theorem,

➨ AC² = AB² + BC²

or

➨ AB² = AC² - BC² = 4k² - 3k² = k²

➨ AB² = k²

Taking square root both side,

➨ AB = k

Finding other T ratios using their definition,

$\rm{ \cos = \frac{AB}{AC} = \frac{1}{2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ \tan = \frac{BC}{AB} = \frac{ \sqrt{ 3 } }{1} } \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ \csc = \frac{1}{ \sin \theta} = \frac{2}{ \sqrt{3} } } \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \rm{ \sec = \frac{1}{ \cos \theta} = \frac{1}{ \frac{1}{2} } = 2 } \: \: \: \: \: \: \: \: \\ \\\rm{ \cot = \frac{1}{ \tan \theta} = \frac{1}{ \frac{ \sqrt{3} }{1} } = \frac{1}{ \sqrt{3} } }$

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1. If 4 sin^2 = 3 . find all other T - ratio.

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Answer 2

$\green{\large\underline{\sf{Solution-}}}$

Given that

$\red{\rm :\longmapsto\:sin \theta \: = \dfrac{ \sqrt{3} }{2} \: }$

can be rewritten as

$\red{\rm :\longmapsto\:sin \theta \: = sin60 \degree \: }$

$\red{\rm \implies\:\theta = 60\degree \: }$

So, other Trigonometric ratios are

$\green{\rm :\longmapsto\:cos60\degree = \dfrac{1}{2} \: }$

$\blue{\rm :\longmapsto\:tan60\degree = \sqrt{3} \: }$

$\purple{\rm :\longmapsto\:cosec60\degree = \dfrac{2}{ \sqrt{3} } \: }$

$\pink{\rm :\longmapsto\:sec60\degree = 2 \: }$

$\orange{\rm :\longmapsto\:cot60\degree = \dfrac{1}{ \sqrt{3} } \: }$

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$\blue{\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}}$