What is the value of cos225+sin165
To find the value of cos(225) and sin(165), you can use the unit circle and trigonometric identities.
First, let's find the value of cos(225):
1. Convert 225 degrees to radians:
- To convert degrees to radians, multiply the degree measure by π/180.
- So, 225 degrees in radians is: 225 * (π/180) = (5π/4) radians.
2. Look at the unit circle. The cosine of an angle is the x-coordinate of the point on the unit circle corresponding to that angle.
- For the angle (5π/4) radians, the x-coordinate is -√2/2.
- Thus, cos(225) = -√2/2.
Next, let's find the value of sin(165):
1. Convert 165 degrees to radians:
- Similarly, multiply 165 by π/180 to get (11π/12) radians.
2. The sine of an angle is the y-coordinate of the point on the unit circle corresponding to that angle.
- For the angle (11π/12) radians, the y-coordinate is √3/2 - 1/2.
- Thus, sin(165) = √3/2 - 1/2.
Now, to find the value of cos(225) + sin(165), substitute the values we found above:
cos(225) + sin(165) = (-√2/2) + (√3/2 - 1/2)
= (√3/2 - √2/2 - 1/2).
Therefore, the value of cos(225) + sin(165) is (√3/2 - √2/2 - 1/2).
that would be the same as
-cos45 + sin15
= -1/√2 + (√3-1)/2√2 = (-2+√3-1)/(2√2) = 1/2 (√3-3)/√2 = 1/4 (√6-3√2)