Use the appropriate sum/difference formula to find the exact value of cos(105°).
do we choose one, then make 1 side equal cos of 105°? I don't understand the question.
105 ° = 210 ° / 2
210 ° = 180 ° + 30 °
cos ( A + B ) = cos ( A ) * cos ( B ) - sin ( A ) * sin ( B )
sin 30 ° = 1 / 2
sin 180 ° = 0
cos 30 ° = sqrt ( 3 ) / 2
cos 180 ° = - 1
cos 210 ° = cos ( 180 ° + 30 ° ) = cos 180 ° * cos 30 ° - sin 180 ° * sin 30 ° =
( - 1 ) * sqrt ( 3 ) / 2 - 0 * 1 / 2 =
- sqrt ( 3 ) / 2 - 0
cos 210 ° = - sqrt ( 3 ) / 2
cos ( theta / 2 ) = ± sqrt [ ( 1 / 2 ) * ( 1 + cos ( theta )) ]
cos 105 ° = cos ( 210 ° / 2 ) =
± sqrt [ ( 1 / 2 ) * ( 1 + ( - sqrt ( 3 ) / 2 ) ] =
± sqrt [ ( 1 / 2 ) * ( 2 / 2 + ( - sqrt ( 3 ) / 2 ) ] =
± sqrt [ ( 1 / 2 ) * ( ( 2 - sqrt ( 3 ) / 2 ) ) ] =
± sqrt [ ( ( 2 - sqrt ( 3 ) ) / 4 ] =
± ( 1 / 2 ) sqrt [ 2 - sqrt ( 3 ) ] =
± sqrt [ 2 - sqrt ( 3 ) ] / 2
The angle 105 ° is in the II quadrant
In quadrant II cosine is negative.
So
cos ( 105 ° ) = - sqrt [ 2 - sqrt ( 3 ) ] / 2
To find the exact value of cos(105°), we can use the difference formula for cosine, which states:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
So, in this case, we can rewrite cos(105°) as cos(45° + 60°):
cos(105°) = cos(45° + 60°)
Now, we can use the difference formula:
cos(105°) = cos(45°)cos(60°) + sin(45°)sin(60°)
We know that cos(45°) = sin(45°) = √2/2 and cos(60°) = 0.5 and sin(60°) = √3/2.
Substituting these values into the formula, we get:
cos(105°) = (√2/2)(0.5) + (√2/2)(√3/2)
Simplifying further:
cos(105°) = (√2/4) + ((√2)(√3))/(2*2)
cos(105°) = (√2/4) + (√6/4)
Finally, combining the two terms:
cos(105°) = (√2 + √6)/4
So, the exact value of cos(105°) is (√2 + √6)/4.
To find the exact value of cos(105°) using the appropriate sum/difference formula, we can utilize the formula for cosine of the difference of two angles:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
In this case, we can choose A = 120° and B = 15°, because their sum is equal to the given angle of 105°:
A + B = 120° + 15° = 135°.
Now, we substitute these values into the formula:
cos(105°) = cos(120° - 15°)
Using the formula:
cos(120° - 15°) = cos(120°)cos(15°) + sin(120°)sin(15°)
Now, we need to find the exact values of cos(120°), cos(15°), sin(120°), and sin(15°).
We can use the unit circle to determine these values:
cos(120°) is the x-coordinate at 120° on the unit circle, which is -1/2.
cos(15°) is the x-coordinate at 15° on the unit circle, which is (√6 + √2)/4.
sin(120°) is the y-coordinate at 120° on the unit circle, which is -√3/2.
sin(15°) is the y-coordinate at 15° on the unit circle, which is (√6 - √2)/4.
Now, we plug in these values into the equation:
cos(120° - 15°) = (-1/2)((√6 + √2)/4) + (-√3/2)((√6 - √2)/4)
Simplifying this expression will give us the exact value of cos(105°).