in triangle ABC,angle b is 90 degree.If sinA =3/4,then show that cos A +sin C =√7/2
Answers
Step-by-step explanation:
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cos A + sin C = √7/2
Given :
Angle B = 90°
sin A = 3/4
To prove :
cos A + sin C = √7/2
Solution :
We know;
sine = opposite side / hypotenuse
cosine = adjacent side / hypotenuse
In the mentioned right-angled triangle ABC, since angle B = 90°,
AC is the hypotenuse(side opposite to angle B)
It is given that sin A = 3/4
⇒ sin A = side opposite to angle A / hypotenuse
⇒ sin A = BC / AC = 3/4
⇒ BC = 3 and AC = 4
By Pythagoras theorem,
AB = √((AC)² - (BC)²) = √(4²-3²) = √(16 - 9) = √7
⇒ AB = √7
cos A = side adjacent to angle A / hypotenuse = AB/AC
⇒ cos A = √7/4
sin C = side opposite to angle C / hypotenuse = AB/AC
⇒ sin C = √7/4
Therefore cos A + sin C = √7/4 + √7/4 = √7/2
Hence proved, cos A + sin C = √7/2
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