Question

Q14. What is the value of TanX , if cos X =1/2

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Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:\tan\:x\:=\:\sqrt{3}}}}$

Step-by-step-explanation:

We have given that, $\displaystyle{\sf\:\cos\:x\:=\:\dfrac{1}{2}}$

We have to find the value of $\displaystyle{\sf\:\tan\:x}$

Now,

$\displaystyle{\sf\:\cos\:x\:=\:\dfrac{1}{2}}$

We know that,

$\displaystyle{\pink{\sf\:\tan\:x\:=\:\dfrac{\sin\:x}{\cos\:x}}\sf\:\:\:-\:-\:-\:[\:Trigonometric\:identity\:]}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\left(\:\dfrac{\sin\:x}{\cos\:x}\:\right)^2\:\:\:-\:-\:-\:[\:Squaring\:both\:sides\:]}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{\sin^2\:x}{\cos^2\:x}\:\:\:-\:-\:-\:\left[\:\because\:\left(\:\dfrac{a}{b}\:\right)^2\:=\:\dfrac{a^2}{b^2}\:\right]}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{1\:-\:\cos^2\:x}{\cos^2\:x}\:\:\:-\:-\:-\:[\:\because\:\sin^2\:x\:+\:\cos^2\:x\:=\:1\:]}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{1\:-\:\left(\:\dfrac{1}{2}\:\right)^2}{\left(\:\dfrac{1}{2}\:\right)^2}}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{1\:-\:\dfrac{1}{2}\:\times\:\dfrac{1}{2}}{\dfrac{1}{2}\:\times\:\dfrac{1}{2}}}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{1\:-\:\dfrac{1}{4}}{\dfrac{1}{4}}}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{\dfrac{4\:-\:1}{4}}{\dfrac{1}{4}}}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{\dfrac{3}{4}}{\dfrac{1}{4}}}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:\dfrac{3}{\cancel{4}}\:\times\:\dfrac{\cancel{4}}{1}}$

$\displaystyle{\implies\sf\:\tan^2\:x\:=\:3}$

$\displaystyle{\implies\sf\:\tan\:x\:=\:\sqrt{3}\:\:\:-\:-\:-\:[\:Taking\:square\:roots\:]}$

$\displaystyle{\therefore\:\underline{\boxed{\red{\sf\:\tan\:x\:=\:\sqrt{3}}}}}$

Answer 2

Step-by-step explanation:

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