Evaluate without using a calculator or tables
sin(pi/3 - 3pi/4)
The answer is - 1+ square root of 3 / 2 square root of 2
How do I get this answer?
If it makes it any easier for you to see the angles in degrees, convert them
pi/3 radians = 60º and 3pi/4 = 135
so sin(pi/3-3pi/4) = sin(60-135) = sin
(-75)
now sin(-75) = -sin(75)
-sin(75) = -sin(45+30)
use the formula we used in the previous posted like questions.
I will check your answer.
To evaluate the expression sin(pi/3 - 3pi/4) without using a calculator or tables, we can use the trigonometric identities.
The given expression involves subtraction of angles in the sine function. We can use the formula sin(A - B) = sin(A)cos(B) - cos(A)sin(B).
In this case, A = pi/3 and B = 3pi/4. Substituting the values into the formula, we get:
sin(pi/3 - 3pi/4) = sin(pi/3)cos(3pi/4) - cos(pi/3)sin(3pi/4)
To evaluate cos(3pi/4) and sin(3pi/4), we can use other trigonometric identities:
cos(3pi/4) = cos(pi/2 - pi/4) = cos(pi/2)cos(pi/4) + sin(pi/2)sin(pi/4)
= 0 * square root of 2 / 2 + 1 * square root of 2 / 2 = square root of 2 / 2
sin(3pi/4) = sin(pi/2 - pi/4) = sin(pi/2)cos(pi/4) - cos(pi/2)sin(pi/4)
= 1 * square root of 2 / 2 - 0 * square root of 2 / 2 = square root of 2 / 2
Now, we can substitute the values back into the original expression:
sin(pi/3 - 3pi/4) = sin(pi/3) * square root of 2 / 2 - cos(pi/3) * square root of 2 / 2
= (square root of 3 / 2) * (square root of 2 / 2) - (1/2) * (square root of 2 / 2)
= (square root of 3 * square root of 2) / 4 - (square root of 2) / 4
= (square root of 3 * square root of 2 - square root of 2) / 4
= (square root of 6 - square root of 2) / 4
Therefore, the final answer is: (square root of 6 - square root of 2) / 4, which can also be expressed as (-1 + square root of 3) / (2 * square root of 2).