Find the exact value by using an appropriate sum or difference identity.
cos(165) degrees
To find the exact value of cos(165) degrees, we can use the sum or difference identity for cosine. The cosine function has an identity known as the cosine of a difference formula, which states that:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
In this case, we can use the identity by expressing 165 degrees as a difference between two angles. We know that cos(180 - 15) degrees = cos(165) degrees.
Now, we can substitute A = 180 and B = 15 into the formula:
cos(165) = cos(180 - 15) = cos(180)cos(15) + sin(180)sin(15)
To evaluate cos(180) and sin(180), we need to know their exact values. In trigonometry, we know that cos(180) = -1 and sin(180) = 0.
Substituting these values into the formula, we get:
cos(165) = (-1)cos(15) + (0)sin(15) = -cos(15)
Now, the remaining step is to find the value of cos(15) degrees. We can use another identity called the half-angle identity for cosine:
cos(2θ) = 2cos^2(θ) - 1
In this case, θ = 15 degrees, so we have:
cos(30) = 2cos^2(15) - 1
To solve for cos(15), we can rearrange the equation:
2cos^2(15) = cos(30) + 1
cos^2(15) = (cos(30) + 1)/2
To find the square root of both sides, we get:
cos(15) = ±√[(cos(30) + 1)/2]
However, since 0 < 15 < 90, and the cosine function is positive in the first and fourth quadrants, we take the positive square root:
cos(15) = √[(cos(30) + 1)/2]
Now, let's find the value of cos(30) using the special triangles or a reference angle in the first quadrant.
In a 30-60-90 degree triangle, the cosine of 30 degrees is √3 / 2.
Substituting this value, we have:
cos(15) = √[(√3 / 2 + 1)/2]
Simplifying further:
cos(15) = √[(√3 + 2)/4]
Therefore, the exact value of cos(165) degrees is √[(√3 + 2)/4].
For these type of questions, you must split up your given angles into a sum or a difference of angles whose trig ratios you know
e.g. 165 = 120+45
then cos(165)
= cos(120+45)
= cos120cos45 - sin120sin45
you should know both sin45 and cos45
for sin120,
sin120 = sin60
cos120 = -cos60
take it from here