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.