Cos 315°
Cos 930°
315° = 360° - 45°
cos 315° = cos ( 360° - 45° )
cos ( A - B ) = sin A * sin B + cos A * cos B
In this case: A = 360° , B = 45°
sin A = sin 360° = 0
cos A = 1
sin B = sin 45° = sqroot ( 2 ) / 2
cos B = cos 45° = sqroot ( 2 ) / 2
cos ( A - B ) = cos ( 360° - 45° ) = sin A * sin B + cos A * cos B
cos ( 360° - 45° ) = 0 * sqroot ( 2 ) / 2 + 1 * sqroot ( 2 ) / 2 =
0 + sqroot ( 2 ) / 2 = sqroot ( 2 ) / 2
cos 315° = sqroot ( 2 ) / 2
You can write this like:
cos 315° = sqroot ( 2 ) / 2 =
sqroot ( 2 ) / sqroot ( 2 ) * sqroot ( 2 ) =
1 / sqroot ( 2 )
So:
cos 315° = sqroot ( 2 ) / 2 = 1 / sqroot ( 2 )
To find the value of cos(315°), we can use the fact that cosine is a periodic function with a period of 360°. This means that cos(315°) is the same as cos(315° - 360°).
Since 315° - 360° = -45°, we need to find the value of cos(-45°).
To find the cosine of -45°, we can use the unit circle or the trigonometric identity for the cosine function.
Considering the unit circle, we know that -45° is in the third quadrant. In the third quadrant, the cosine function is negative, so cos(-45°) = -cos(45°).
Alternatively, we can use the trigonometric identity cos(-θ) = cos(θ) to find that cos(-45°) = cos(45°).
For the value of cos(45°), we can use the fact that it is a special angle for which we know the exact value. In the case of 45°, it is a 45-45-90 right triangle where the adjacent side is equal to the opposite side.
In this triangle, the adjacent side is equal to the opposite side, and the hypotenuse is the square root of 2 times the length of the sides.
Therefore, cos(45°) = adjacent side / hypotenuse = (1 / √2) = √2 / 2.
So, cos(315°) = -cos(45°) = -√2 / 2.