Question

In a ABC, it is given that B  90, AB = 7cm and ( AC – BC ) = 1cm. Find the values of sinA , cosA , sin C and cos C .

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:\sin\:A\:=\:\dfrac{24}{25}\:}}}$

$\displaystyle{\boxed{\green{\sf\:\cos\:A\:=\:\dfrac{7}{25}\:}}}$

$\displaystyle{\boxed{\blue{\sf\:\sin\:C\:=\:\dfrac{7}{25}\:}}}$

$\displaystyle{\boxed{\orange{\sf\:\cos\:C\:=\:\dfrac{24}{25}\:}}}$

Step-by-step-explanation:

NOTE: Refer to the attachment for the diagram.

We have given that,

In figure, in △ABC,

• m∠B = 90°
• AB = 7 cm
• ( AC - BC ) = 1 cm

We have to find the the value of sin A, cos A, sin C and cos C.

Now,

AC - BC = 1

⇒ AC = 1 + BC $\sf\qquad\cdots\:(\:1\:)$

Now, in △ABC, m∠B = 90°

∴ By Pythagoras theorem,

( AC )² = ( AB )² + ( BC )²

⇒ ( AC )² - ( BC )² = ( AB )²

⇒ ( AC - BC ) ( AC + BC ) = ( 7 )²

⇒ ( 1 + BC - BC ) ( 1 + BC + BC ) = 49 $\sf\qquad\cdots\:[\:From\:(\:1\:)\:]$

⇒ 1 * ( 1 + 2BC ) = 49

⇒ 2BC + 1 = 49

⇒ 2BC = 49 - 1

⇒ 2BC = 48

⇒ BC = 48 ÷ 2

⇒ BC = 24 cm

By using this value in equation ( 1 ),

AC = 1 + BC $\sf\qquad\cdots\:(\:1\:)$

⇒ AC = 1 + 24

⇒ AC = 25 cm

Now, we know that,

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

$\displaystyle{\implies\sf\:\sin\:A\:=\:\dfrac{BC}{AC}}$

$\displaystyle{\implies\boxed{\red{\sf\:\sin\:A\:=\:\dfrac{24}{25}\:}}}$

Now,

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

$\displaystyle{\implies\sf\:\cos\:A\:=\:\dfrac{AB}{AC}}$

$\displaystyle{\implies\boxed{\green{\sf\:\cos\:A\:=\:\dfrac{7}{25}\:}}}$

Now,

$\displaystyle{\sf\:\sin\:C\:=\:\dfrac{AB}{AC}}$

$\displaystyle{\implies\boxed{\blue{\sf\:\sin\:C\:=\:\dfrac{7}{25}\:}}}$

Now,

$\displaystyle{\sf\:\cos\:C\:=\:\dfrac{BC}{AC}}$

$\displaystyle{\implies\boxed{\orange{\sf\:\cos\:C\:=\:\dfrac{24}{25}\:}}}$