using the trigonometric identities find the exact (radical form) value of cos (105 degrees)
cos 105°
= cos(60+45)
= cos60cos45 - sin60sin45
= (1/2)(√2/2) - (√3/2)(√2/2)
= (√2 - √6)/4
cos195
To find the exact value of cos(105 degrees) using trigonometric identities, we can make use of the sum-of-angles identity, specifically the one for cosines, which states:
cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
In this case, we can rewrite 105 degrees as the sum of two angles using the identity: 105 = 45 + 60.
Therefore, cos(105 degrees) = cos(45 degrees + 60 degrees)
Using the sum-of-angles identity, we have:
cos(105 degrees) = cos(45 degrees)cos(60 degrees) - sin(45 degrees)sin(60 degrees)
Now, let's determine the values for cos(45 degrees) and sin(45 degrees), which are well-known values:
cos(45 degrees) = √(2)/2
sin(45 degrees) = √(2)/2
Next, let's determine the values for cos(60 degrees) and sin(60 degrees):
cos(60 degrees) = 1/2
sin(60 degrees) = √(3)/2
Plugging in these values into the equation, we have:
cos(105 degrees) = (√2/2)(1/2) - (√2/2)(√3)/2
Simplifying this expression further:
cos(105 degrees) = (√2)/(4) - (√6)/(4) = (√2 - √6)/(4)
Hence, the exact value of cos(105 degrees) in radical form is (√2 - √6)/4.