Evaluate the expression without using a calculator(show work please)?
(sin pi/4 cos pi/6 - sin pi/6 cos pi/4)
the answer is 1/4(2-�ã3)
To evaluate the expression (sin(pi/4) * cos(pi/6) - sin(pi/6) * cos(pi/4)) without using a calculator, we can use the trigonometric identities and simplify the expression step by step.
1. First, recall the values of sine and cosine for specific angles:
- sin(pi/4) = √2/2
- cos(pi/6) = √3/2
- sin(pi/6) = 1/2
- cos(pi/4) = √2/2
2. Substitute the values into the expression:
(sin(pi/4) * cos(pi/6) - sin(pi/6) * cos(pi/4))
= (√2/2 * √3/2) - (1/2 * √2/2)
3. Simplify each term individually:
- (√2/2 * √3/2) = (√6/4)
(To simplify the multiplication, multiply the numerators and the denominators separately.)
- (1/2 * √2/2) = (√2/4)
(Similar to the step above, multiply the numerators and the denominators.)
4. Now re-write the expression:
= (√6/4) - (√2/4)
5. Combine the terms together:
= (√6 - √2) / 4
6. The final step is to multiply by 1/4 to get the common denominator:
= 1/4(√6 - √2)
Thus, the evaluated expression is 1/4(2 - √3)