Math, asked by samitp1983, 5 hours ago

Find cos a and sin B

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Answers

Answered by varadad25

Answer:

$\displaystyle{\boxed{\red{\sf\:\cos\:\alpha\:=\:\dfrac{3}{5}\:}}}$

$\displaystyle{\boxed{\blue{\sf\:\sin\:\beta\:=\:\dfrac{20}{29}\:}}}$

Step-by-step-explanation:

In figure, in △PQR,

m∠PQR = 90°

∴ By Pythagoras theorem,

( PR )² = ( PQ )² + ( QR )²

⇒ ( 20 )² = ( 16 )² + ( QR )²

⇒ ( QR )² = ( 20 )² - ( 16 )²

⇒ ( QR )² = ( 20 + 16 ) ( 20 - 16 ) $\displaystyle{\qquad\dots}$ [ a² - b² = ( a + b ) ( a - b ) ]

⇒ ( QR )² = 36 * 4

⇒ QR = √( 36 * 4 )

⇒ QR = 6 * 2

⇒ QR = 12

Now, we know that,

$\displaystyle{\pink{\sf\:\cos\:\alpha\:=\:\dfrac{Adjacent\:side}{Hypotenuse}}}$

$\displaystyle{\implies\sf\:\cos\:\alpha\:=\:\dfrac{QR}{PR}}$

$\displaystyle{\implies\sf\:\cos\:\alpha\:=\:\dfrac{\cancel{12}}{\cancel{20}}}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:\cos\:\alpha\:=\:\dfrac{3}{5}\:}}}}$

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Now, in △PRS,

m∠PRS = 90°

∴ By Pythagoras theorem,

( PS )² = ( PR )² + ( RS )²

⇒ ( PS )² = ( 20 )² + ( 21 )²

⇒ ( PS )² = 400 + 441

⇒ ( PS )² = 841

⇒ PS = √841

⇒ PS = √( 29 * 29 )

⇒ PS = 29

Now,

$\displaystyle{\pink{\sf\:\sin\:\beta\:=\:\dfrac{Opposite\:side}{Hypotenuse}}}$

$\displaystyle{\implies\sf\:\sin\:\beta\:=\:\dfrac{PR}{PS}}$

$\displaystyle{\implies\underline{\boxed{\blue{\sf\:\sin\:\beta\:=\:\dfrac{20}{29}\:}}}}$