give exact value of cos 75 degrees.
give exact value sin 15 degrees.
give exact value sin 225 degrees.
You will have to use the addition relations here. Notice 75 is 30 +45, 15 is 45-30, and 225 is 180 + 45.
I will be happy to critique your work.
how would you find csc^2 of 42 degrees then?
To find the exact value of cos 75 degrees, we will use the addition formula for cosine:
cos(a+b) = cos(a)cos(b) - sin(a)sin(b)
Since we know that 75 degrees is equal to 30 degrees + 45 degrees, we can write:
cos 75 = cos (30 + 45)
Using the addition formula, we can substitute a = 30 and b = 45:
cos (30 + 45) = cos 30 cos 45 - sin 30 sin 45
The exact values of cos 30, cos 45, sin 30, and sin 45 can be derived from special right triangles or trigonometric identities:
cos 30 = √3/2
cos 45 = √2/2
sin 30 = 1/2
sin 45 = √2/2
Substituting these values, we get:
cos 75 = (√3/2)(√2/2) - (1/2)(√2/2)
= (√6 + √2)/4
Therefore, the exact value of cos 75 degrees is (√6 + √2)/4.
For sin 15 degrees, we will use the subtraction formula for sine:
sin(a-b) = sin(a)cos(b) - cos(a)sin(b)
Since 15 degrees is equal to 45 degrees - 30 degrees, we can write:
sin 15 = sin (45 - 30)
Using the subtraction formula, we can substitute a = 45 and b = 30:
sin (45 - 30) = sin 45 cos 30 - cos 45 sin 30
Using the same exact values as before:
sin 15 = (√2/2)(√3/2) - (√2/2)(1/2)
= (√6 - √2)/4
Therefore, the exact value of sin 15 degrees is (√6 - √2)/4.
Lastly, for sin 225 degrees, we can rewrite it as sin (180 + 45) using the addition formula.
sin 225 = sin (180 + 45)
Since sin (180 + θ) = -sin θ, we have:
sin (180 + 45) = - sin 45
Using the value of sin 45 from before (sin 45 = √2/2):
sin 225 = - (√2/2)
= -√2/2
Therefore, the exact value of sin 225 degrees is -√2/2.