evaluate: cos (arc sin (4/5) - arc tan 2) with out using a calc.
cos (arc sin (4/5) - arc tan 2)
remember that arcsin(4/5) and arctan2 are just angles,
so
cos (arc sin (4/5) - arc tan 2)
= cos(arcsin(4/5)cos(arctan2) + sin(arcsin(4/5))sin(arctan2))
= (3/5)(1/√5) + (4/5)(2/√5) = 11/(5√5)
To evaluate the expression cos(arc sin(4/5) - arc tan(2) ) without using a calculator, we can use the trigonometric identities and properties.
1. Start with the expression: cos(arc sin(4/5) - arc tan(2) ).
2. Use the identity for the difference of angles of the cosine function: cos(A - B) = cos(A)cos(B) + sin(A)sin(B).
3. Identify A = arc sin(4/5) and B = arc tan(2). Rewrite the expression using these
identities: cos(arc sin(4/5))cos(arc tan(2)) + sin(arc sin(4/5))sin(arc tan(2)).
4. Use the identity for the sine and cosine of arcsin(x): sin(arc sin(x)) = x and cos(arc sin(x)) = sqrt(1 - x^2).
Replace sin(arc sin(4/5)) with 4/5 and cos(arc sin(4/5)) with sqrt(1 - (4/5)^2) = 3/5.
5. Use the identity for the sine and cosine of arctan(x): sin(arc tan(x)) = x/sqrt(1 + x^2) and cos(arc tan(x)) = 1/sqrt(1 + x^2).
Replace sin(arc tan(2)) with 2/sqrt(1 + 2^2) = 2/sqrt(5) and cos(arc tan(2)) with 1/sqrt(1 + 2^2) = 1/sqrt(5).
6. Replace cos(arc sin(x)) with 3/5 and cos(arc tan(x)) with 1/sqrt(5) in the expression:
(3/5)(1/sqrt(5)) + (4/5)(2/sqrt(5)).
7. Simplify the expression:
(3/5)(1/sqrt(5)) + (4/5)(2/sqrt(5)) = 3/(5sqrt(5)) + 8/(5sqrt(5)) = (3 + 8)/(5sqrt(5)) = 11/(5sqrt(5)).
So, the simplified form of cos(arc sin(4/5) - arc tan(2)) is 11/(5sqrt(5)).