Use the Sum and Difference properties to find the exact value of
sin 105°
sin(60+45) = sin60 cos45 + cos60 sin45
Now plug in those familiar values.
If you knew to use the sum formula, why didn't you just use it?
To find the exact value of sin 105° using the Sum and Difference properties, we need to express 105° as a sum or difference of angles whose sin values we know.
We can express 105° as the sum of two angles, 45° and 60°. So, we can use the formula sin(A + B) = sin A * cos B + cos A * sin B.
Let's express 105° as the sum of 45° and 60°:
105° = 45° + 60°
Now, we can apply the Sum and Difference properties:
sin 105° = sin(45° + 60°)
= sin 45° * cos 60° + cos 45° * sin 60°
We know the exact values of sin 45° and sin 60°, so let's substitute them in:
sin 105° = (sqrt(2)/2) * (1/2) + (sqrt(2)/2) * (sqrt(3)/2)
Evaluating the expression:
sin 105° = sqrt(2)/4 + sqrt(2)/2 * sqrt(3)/2
= sqrt(2)/4 + sqrt(6)/4
Since the denominators are the same, we can combine the numerators:
sin 105° = (sqrt(2) + sqrt(6))/4
Therefore, the exact value of sin 105° using the Sum and Difference properties is (sqrt(2) + sqrt(6))/4.