Find the exact value of: sin(7pi/6 + 3pi/4)

Question

Find the exact value of: sin(7pi/6 + 3pi/4)

Answer 1

For this problem, you'll want to use the Trigonometric Addition Formula for sine:

 

sin(x+y) = sin(x)cos(y) + sin(y)cos(x)

 

 

 

Let (7π)/6 = x and let (3π)/4 = y, and plug those values into the equation:

 

sin(x+y) = sin(x)cos(y) + sin(y)cos(x)

sin(7π/6 + 3π/4) = sin(7π/6)cos(3π/4) + sin(3π/4)cos(7π/6)

 

 

 

Now, let's figure out the sines and cosines of those two angles.

 

 

7π/6 means we've gone π/6 past halfway around the unit circle (6π/6). If it helps you picture it, the angle in degrees is 210°. It's in the Third Quadrant, so sine and cosine are negative. That means that:

 

sin(7π/6) = -1/2

cos(7π/6) = -√(3)/2

 

 

3π/4 means we've gone π/4 past a quarter of the way around the unit circle (2π/4). If it helps you picture it, the angle in degrees is 135°. It's in the Second Quadrant, so sine is positive and cosine is negative. That means that:

 

sin(3π/4) = √(2)/2

cos(3π/4) = -√(2)/2

 

 

 

Plug those values into the equation and solve:

 

sin(7π/6 + 3π/4) = sin(7π/6)cos(3π/4) + sin(3π/4)cos(7π/6)

sin(7π/6 + 3π/4) = (-1/2)(-√(2)/2) + (√(2)/2)(-√(3)/2)

sin(7π/6 + 3π/4) = (√(2)/4) + (-√(6)/4)

sin(7π/6 + 3π/4) = (√(2) - √(6)) / 4