Use a sum or difference identity to find the exact value of sin 255 degrees
Answer: sqrt6-sqrt2/4
Thanks
You might notice that 225 = 180 + 75
so sin 255 = sin(180+75)
= sin 180cos 75 + cos180sin75
= -sin75
now 75 = 45 + 30
so sin(75) = sin(45+30)
= sin45cos30 + cos45sin30
= (1/√2)(√3/2) + (1/√2)(1/2) or
= (√2/2)(√3/2) + )√2/2)(1/2)
= (√6 + √2)/4
so sin 255 = -(√6 + √2)/4
To find the exact value of sin 255 degrees using a sum or difference identity, we need to express 255 degrees as the sum or difference of two angles with known sine values.
First, let's find an angle that is a multiple of 30 degrees. Since 255 = 8 * 30 + 15, we can express 255 degrees as the sum of 240 degrees and 15 degrees.
Next, we can use the sine difference identity:
sin(A - B) = sinAcosB - cosAsinB
Let's use this identity to find sin 255 degrees:
sin 255 degrees = sin(240 degrees + 15 degrees)
Applying the sine difference identity:
sin(240 degrees + 15 degrees) = sin 240 degrees * cos 15 degrees - cos 240 degrees * sin 15 degrees
Now, we need to find the sine and cosine values for 240 degrees and 15 degrees. These values can be determined from the unit circle or trigonometric identities.
From the unit circle:
- sin 240 degrees = -√3/2
- cos 240 degrees = -1/2
- sin 15 degrees = √6 - √2 / 4
- cos 15 degrees = √6 + √2 / 4
Substituting these values into the previous equation:
sin 255 degrees = (-√3/2) * (√6 + √2 / 4) - (-1/2) * (√6 - √2 / 4)
To simplify further, we can multiply the terms and rationalize the denominators:
sin 255 degrees = -√(18) / 8 - √(6)/ 8 + √(6) / 8 + √(2) / 8
Combining like terms:
sin 255 degrees = -√(18) / 8 + √(6) / 8 - √(6) / 8 + √(2) / 8
Now, combining the terms with common denominators, we get:
sin 255 degrees = (-√(18) + √(2)) / 8
Further simplifying, we can rationalize the numerator by multiplying both the numerator and denominator by √2:
sin 255 degrees = (-√(18) + √(2)) / 8 * (√2 / √2)
This gives us:
sin 255 degrees = (-√(18)√2 + √(2)√2) / (8√2)
Simplifying the numerator:
sin 255 degrees = (-√36 + √2) / (8√2)
Since √36 = 6, the expression becomes:
sin 255 degrees = (-6 + √2) / (8√2)
Finally, we can rationalize the denominator by multiplying both the numerator and denominator by √2:
sin 255 degrees = (-6√2 + √2√2) / (8 * 2)
This simplifies to:
sin 255 degrees = (-6√2 + √4) / 16
Finally, simplifying the expression:
sin 255 degrees = -6√2 + 2) / 16
This gives us the exact value of sin 255 degrees as:
sqrt6-sqrt2/4
To find the value of sin 255 degrees using a sum or difference identity, we can use the identity for sine of a difference:
sin(A - B) = sin A cos B - cos A sin B.
First, we need to find two angles whose sine values we know, which add up to 255 degrees.
Let's express 255 degrees as a sum or difference of angles whose sine values we know:
255 degrees = 225 degrees + 30 degrees.
Now, we can apply the sine difference identity:
sin(255 degrees) = sin(225 degrees + 30 degrees)
= sin(225 degrees) cos(30 degrees) - cos(225 degrees) sin(30 degrees).
To find the values of sin(225 degrees) and cos(225 degrees), we can use the fact that sin and cos are periodic functions:
sin(225 degrees) = sin(225 degrees - 360 degrees) = sin(-135 degrees)
= -sin(135 degrees) = -sin(135 degrees - 360 degrees)
= -sin(-225 degrees) = -(-sin(225 degrees)) = sin(225 degrees),
cos(225 degrees) = cos(225 degrees - 360 degrees) = cos(-135 degrees)
= cos(135 degrees) = -cos(135 degrees - 360 degrees)
= -cos(-225 degrees) = -(-cos(225 degrees)) = cos(225 degrees).
Now, substituting these values into the equation:
sin(255 degrees) = sin(225 degrees) cos(30 degrees) - cos(225 degrees) sin(30 degrees)
= sin(225 degrees) * (sqrt(3)/2) - cos(225 degrees) * (1/2)
= sin(225 degrees) * sqrt(3)/2 - cos(225 degrees)/2.
Since sin(225 degrees) is equal to sin(225 degrees), we can replace it in the equation:
sin(255 degrees) = sin(225 degrees) * sqrt(3)/2 - cos(225 degrees)/2
= sin(225 degrees) * sqrt(3)/2 - cos(225 degrees)/2.
Now, we just need to find the value of sin(225 degrees) to determine the exact value of sin(255 degrees).
Using the unit circle or reference angles, we know that sin(225 degrees) = -sqrt(2)/2.
Substituting this value into the equation:
sin(255 degrees) = sin(225 degrees) * sqrt(3)/2 - cos(225 degrees)/2
= (-sqrt(2)/2) * sqrt(3)/2 - cos(225 degrees)/2
= -sqrt(6)/4 + cos(225 degrees)/2.
To find the value of cos(225 degrees), we can use the fact that cos is an even function:
cos(225 degrees) = cos(-225 degrees) = cos(225 degrees - 360 degrees)
= cos(-135 degrees)
= cos(135 degrees) = -cos(135 degrees - 360 degrees) = -cos(-225 degrees)
= -cos(225 degrees).
Now, substituting this value into the equation:
sin(255 degrees) = -sqrt(6)/4 + cos(225 degrees)/2
= -sqrt(6)/4 + (-cos(225 degrees))/2
= -sqrt(6)/4 - cos(225 degrees)/2.
So, the exact value of sin 255 degrees is -sqrt(6)/4 - cos(225 degrees)/2, which simplifies to sqrt(6)-sqrt(2)/4.
Therefore, the exact value of sin 255 degrees is sqrt(6)-sqrt(2)/4.