Find sin (alpha/2) and cos (alpha/2)

sin alpha = 3, -pi/2 < alpha <less then or equal to 0

How to get the answer of sin (alpha/2) = -1/square root 3, cos(alpha/2) = square root 6/3?

You said "sin alpha = 3"

check your typing, the sine of any angle must be between ±1

Looking at your answers, α/2 is in the fourth quadrant, with x=√2, y=-1, and r=√3

To find the values of sin (alpha/2) and cos (alpha/2), we can use the half-angle formula.

1. Start with the given information: sin(alpha) = 3 and -pi/2 < alpha <= 0.

2. Since the value of sin(alpha) is positive, we know that alpha must lie in the second quadrant (-pi/2 < alpha < pi/2).

3. Now, we can use the half-angle formula for sine: sin(alpha/2) = ±√((1 - cos(alpha)) / 2).

4. Substitute sin(alpha) = 3 into the formula: sin(alpha/2) = ±√((1 - cos(alpha)) / 2) = ±√((1 - cos(2pi - alpha)) / 2).

5. Since alpha lies in the second quadrant, alpha is between -90 degrees and 0 degrees. Also, cos(alpha) will be negative because it is in the second quadrant.

6. Let's calculate cos(alpha): We know that sin^2(alpha) + cos^2(alpha) = 1. Since sin(alpha) = 3, we have 9 + cos^2(alpha) = 1. Solving this equation, we get cos(alpha) = -√8.

7. Substitute the value of cos(alpha) into the half-angle formula: sin(alpha/2) = ±√((1 - cos(2pi - alpha)) / 2) = ±√((1 - cos(alpha)) / 2) = ±√((1 - (-√8)) / 2) = ±√((1 + √8) / 2) = ±√(1/2 + √8/2) = ±√(1/2 + √2).

8. Simplify the expression: Using the conjugate, we can rationalize the denominator: ±√(1/2 + √2) × (√2 - 1)/(√2 - 1) = ±(√2 - 1)/√4 - √2 = ±(√2 - 1)/(2 - √2).

9. Multiply the numerator and denominator by the conjugate of the denominator to rationalize it further: (√2 - 1)(2 + √2)/(2 - √2)(2 + √2) = (√2 - 1)(2 + √2)/(4 - 2).

10. Simplify the expression: (√2 - 1)(2 + √2)/(4 - 2) = (√2 - 1)(√2 + 1)/2 = (2 - 1)/2 = 1/2.

Therefore, sin(alpha/2) = ±1/2.

11. To find cos(alpha/2), we can use the half-angle formula for cosine: cos(alpha/2) = ±√((1 + cos(alpha)) / 2).

12. Substitute the value of cos(alpha) into the formula: cos(alpha/2) = ±√((1 + cos(alpha)) / 2) = ±√((1 - √8) / 2) = ±√(1/2 - √8/2) = ±√(1/2 - √2).

13. Simplify the expression: Using the conjugate, we can rationalize the denominator: ±√(1/2 - √2) × (√2 + 1)/(√2 + 1) = ±(√2 + 1)/√4 - 2 = ±(√2 + 1)/(2 - √2).

14. Multiply the numerator and denominator by the conjugate of the denominator to rationalize it further: (√2 + 1)(2 + √2)/(2 - √2)(2 + √2) = (√2 + 1)(2 + √2)/(4 - 2).

15. Simplify the expression: (√2 + 1)(2 + √2)/(4 - 2) = (√2 + 1)(√2 + 1)/2 = (2 + 1)/2 = 3/2.

Therefore, cos(alpha/2) = ±3/2.

In conclusion, sin(alpha/2) is equal to ±1/2, and cos(alpha/2) is equal to ±3/2.