Find the exact value of the expression.
sin(arccos sqrt(5)/5)
remember Sin^2Theta+cos^2Theta=1
so if cosTheta= sqrt5/5, you can find sinTheta.
To find the exact value of the expression sin(arccos(sqrt(5)/5)), we can use the trigonometric identity relating the sine and the cosine of an angle. The identity is as follows:
sin(arccos(x)) = sqrt(1 - x^2)
First, let's find the value of x to substitute into the identity. In this case, x is sqrt(5)/5.
Now substituting into the identity, we get:
sin(arccos(sqrt(5)/5)) = sqrt(1 - (sqrt(5)/5)^2)
To simplify further, we square sqrt(5)/5:
= sqrt(1 - 5/25)
= sqrt(1 - 1/5)
= sqrt(4/5)
= 2/sqrt(5)
To rationalize the denominator, we multiply both the numerator and denominator by sqrt(5):
= (2 * sqrt(5))/(sqrt(5) * sqrt(5))
= 2sqrt(5)/5
Therefore, the exact value of the expression sin(arccos(sqrt(5)/5)) is 2sqrt(5)/5.