find the exact value:
sin(315-60)
To find the exact value of sin(315-60), we can use the trigonometric identity:
sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B)
Let's substitute the values of A = 315 degrees and B = 60 degrees into the formula:
sin(315 - 60) = sin(315) * cos(60) - cos(315) * sin(60)
To evaluate sin(315) and cos(315), we need to express the angles in terms of standard angles or reference angles. Since 315 degrees is equivalent to 45 degrees plus a full revolution (360 degrees), we can express it as:
315 degrees = 360 degrees - 45 degrees
Using this conversion, we can rewrite the expression:
sin(315) * cos(60) - cos(315) * sin(60) = sin(360 - 45) * cos(60) - cos(360 - 45) * sin(60)
Since sin(360 - θ) = sin(θ) and cos(360 - θ) = cos(θ), we can simplify the expression further:
sin(360 - 45) * cos(60) - cos(360 - 45) * sin(60) = sin(45) * cos(60) - cos(45) * sin(60)
Using the values of sin(45) = 1/√2 and cos(45) = 1/√2, and sin(60) = √3/2 and cos(60) = 1/2, we can substitute them into the expression:
sin(45) * cos(60) - cos(45) * sin(60) = (1/√2) * (1/2) - (1/√2) * (√3/2)
simplifying:
= (1/√2) * (1/2) - (1/√2) * (√3/2)
= 1/(2√2) - (√3/(2√2))
= (1 - √3)/(2√2)
Therefore, the exact value of sin(315-60) is (1 - √3)/(2√2).