Question

cosA=x/square root of (x^2+y^2) then prove xsinA=ycosA

Answer 1

Hi there !
Here's the answer :

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¶ POINTS TO REMEMBER :

• $sin = \frac{Opposite}{Hypotenuse}$

• $cos = \frac{Adjacent}{Hypotenuse}$

• Pythagoras theorem :
Hyp² = Opp² + Adj²

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¶¶¶ SOLUTION:

Given,
$cos\: A = \frac{x}{\sqrt{x^{2}+y^{2}}}$ -----------(1)

•°• Adj = x &
Hyp = $x^{2}+y^{2}$

Now, Find Opp using Pythagoras theorem to calculate sin A

Hyp² = Opp² + Adj²

=> Opp² = Hyp² - Adj²

=> Opp² = (x²+y²) - x²

=> Opp² = y²

=> Opp = y

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$sin\: A = \frac{y}{\sqrt{x^{2}+y^{2}}}$ --------(2)

Do (1) ÷ (2)

$\dfrac{cos\: A}{sin\: A}=\dfrac{(\frac{x}{\sqrt{x^{2}+y^{2}}})}{( \frac{y}{\sqrt{x^{2}+y^{2}}})}$

$\implies \dfrac{cos\: A}{sin\: A}= \dfrac{x}{y}$

Cross Multiply

$\implies xsin\: A = ycos\: A$

Hence proved.

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Answer 2

$✅$

the above answer is correct