Question

tell some trigonometry ratios​

Answer 1

Step-by-step explanation:

give way of latex codes

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Integrate the function

$\huge\green\tt\frac{ \sqrt{tanx} }{sinxcosx}}$

⇛$\huge\tt\frac{ \sqrt{tanx} }{sinxcosx}$

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⇛$\huge\tt \frac{ \sqrt{tanx} }{sinxcosx \times \frac{cosx}{cosx}}$

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⇛$\huge\tt \frac{ \sqrt{tanx} }{sinx \times \frac{ {cos}^{2} x}{cosx}}$ ㅤ ㅤ ㅤ

⇛ $\huge\tt\frac{ \sqrt{tanx} }{ {cos}^{2} x \times \frac{sinx}{cosx} }$

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⇛$\huge\tt\frac{ \sqrt{tanx} }{ {cos}^{2}x \times tanx }$

⇛$\huge\tt {tan}^{ \frac{1}{2} - 1 } \times \frac{1}{ {cos}^{2} x}$ㅤ ㅤ ㅤ ㅤ ㅤ

⇛$\huge\tt {(tan)}^{ - \frac{ 1}{2} } \times \frac{1}{ {cos}^{2}x } = {(tanx)}^{ - \frac{1}{2} } \times {sec}^{2} x⇛(tan)$

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⇛$\huge\tt {(tan)}^{ - \frac{ 1}{2} } \times \frac{1}{ {cos}^{2}x } = ∫ {(tanx)}^{ - \frac{1}{2} } \times {sec}^{2} x \times dx⇛(tan)$

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$\bold\blue{☛\: Let tanx=t}$

$\bold\blue{☛ \:Differentiating \: both \: sides \: w.r.t.x}$

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⇛$\huge\tt {sec}^{2} x = \frac{dt}{dx}$

⇛$\huge\tt{dx \frac{dt}{ {sec}^{2}x }}$

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⇛$\huge\tt∴∫ {(tanx)}^{ - \frac{1}{2} } \times {sec}^{2} x \times dx$

⇛$\huge\tt ∫ {(t)}^{ - \frac{1}{2} } \times {sec}^{2} x \times \frac{dt}{ {sec}^{2}x }$

⇛$\huge\tt ∫ {t}^{ - \frac{1}{2} }$ㅤ ㅤ

⇛ $\huge\tt\frac{ {t}^{ - \frac{1}{2} + 1} }{ - \frac{1}{2} + 1 }$

⇛ $\huge\tt \frac{ {t}^{ \frac{1}{2} } }{ \frac{1}{2} } + c = 2 {t}^{ \frac{1}{2} } + c = 2 \sqrt{t}$

⇛$\huge2 \sqrt{t} + c = 2 \sqrt{tanx}$

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Answer 2

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The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others.