Question

Recall the Pythagorean identity sin2(theta) + cos2(theta) = 1, where theta is any real number.

a. Find sin(x), given cos(x) = 35, for − ???? / 2 < x < 0.

Answer 1

-4/5

Explanation:

If ABC is a right triangle having ∠B= 90° and the sides AB, BC, and CA are respectively the perpendicular, the base, and the hypotenuse.

Now, if ∠C is equal to Ф, then Sin Ф =Perpendicular / Hypotenuse = AB/CA and Cos Ф =Base / hypotenuse = BC/CA

Hence, Sin²Ф + Cos²Ф = AB²/CA² + BC²/CA² =(AB²+BC²)/ CA² = CA² /CA² =1 {Using Pythagorean Theorem, AB² +BC² =CA² i.e (Perpendicular)² +(Base)² = (Hypotenuse)²}

Now, if Cosx = 3/5 where -π/2 < x <0 i.e. the value of x lies in 4th quadrant, where Sinx will be negative then

Sin²x= 1-Cos²x =1- (3/5)² =(4/5)²

⇒ Sin x = -4/5 (Answer)