Find the exact value of the trig function below.
cos2Ø if sinØ = 2/5
from sinØ = 2/5, sketch a right-angled triangle
Use Pythagoras to find cos Ø
cos^2 Ø + sin^2 Ø = 1
cos^2 Ø = 21/25
cos 2Ø = cos^2 Ø - sin^2 Ø, you should know that identity
cos 2Ø = 21/25 - 4/25 = 17/25
cos(2x) = 1 - 2sin^2(x)
now just use your numbers.
To find the value of cos(2Ø) given sin(Ø) = 2/5, we can use the Pythagorean identity sin²(Ø) + cos²(Ø) = 1.
First, we need to find the value of cos(Ø). Since we know sin(Ø) = 2/5, we can use the Pythagorean identity to find cos(Ø).
sin²(Ø) + cos²(Ø) = 1
(2/5)² + cos²(Ø) = 1
4/25 + cos²(Ø) = 1
cos²(Ø) = 1 - 4/25
cos²(Ø) = 25/25 - 4/25
cos²(Ø) = 21/25
Taking the square root of both sides, we find:
cos(Ø) = ±√(21/25)
Since sin(Ø) is positive (2/5 is positive) and the cosine function is positive in the first and fourth quadrants, we choose the positive value for cos(Ø):
cos(Ø) = √(21/25)
cos(Ø) = √21/√25
cos(Ø) = √21/5
Now, to find the value of cos(2Ø), we can use the double angle identity:
cos(2Ø) = 2cos²(Ø) - 1
Plugging in the value of cos(Ø):
cos(2Ø) = 2(√21/5)² - 1
cos(2Ø) = 2(21/25) - 1
cos(2Ø) = 42/25 - 1
cos(2Ø) = (42 - 25)/25
cos(2Ø) = 17/25
Therefore, the exact value of cos(2Ø) is 17/25.