Question

Find the exact values of ( sin A ) and ( cos A ) for ( sin A = frac(√(2) - √(3))(√(2) + √(3)) ) on the interval ( 0^circ leq A leq 90^circ ).

a) ( sin A = frac(√(2) - √(3))(√(2) + √(3)), quad cos A = frac(√(2) + √(3))(√(2) + √(3)) )

b) ( sin A = frac(√(2) + √(3))(√(2) - √(3)), quad cos A = frac(√(2) - √(3))(√(2) - √(3)) )

c) ( sin A = frac(√(2) - √(3))(√(2) - √(3)), quad cos A = frac(√(2) + √(3))(√(2) - √(3)) )

d) ( sin A = frac(√(2) + √(3))(√(2) + √(3)), quad cos A = frac(√(2) - √(3))(√(2) + √(3)) )

Answer 1

To find the exact values of sin(A) and cos(A) given sin(A)=(sqrt(2)-sqrt(3))/(sqrt(2)+sqrt(3)), we can rationalize the denominator and simplify the expression. The exact values of sin(A) and cos(A) are (5 - 4sqrt(6))/(sqrt(3) + sqrt(2)) and will be found by using trigonometric identities.

Step-by-step explanation:

In order to find the exact values of sin(A) and cos(A) given sin(A) = (sqrt(2) - sqrt(3))/(sqrt(2) + sqrt(3)), we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

Let's simplify:

Square the conjugate of the denominator: (sqrt(2) + sqrt(3)) * (sqrt(2) - sqrt(3)) = 2 - 3 = -1

Now, multiply the numerator and denominator by (-1):

Sin(A) = (sqrt(2) - sqrt(3)) * (-1) / ((sqrt(2) + sqrt(3)) * (-1))

After canceling out the negative signs, we get:

Sin(A) = (sqrt(3) - sqrt(2)) / (sqrt(3) + sqrt(2))

Now, let's rationalize the denominator one more time:

Multiply the numerator and denominator by the conjugate of the denominator:

(sqrt(3) - sqrt(2)) * (sqrt(3) - sqrt(2)) = 3 - 2sqrt(6) - 2sqrt(6) + 2 = 5 - 4sqrt(6)

Therefore, we have:

Sin(A) = (5 - 4sqrt(6)) / (sqrt(3) + sqrt(2))

To find cos(A), we can use the trigonometric identity cos^2(A) + sin^2(A) = 1. Rearrange this equation to solve for cos(A):

Cos^2(A) = 1 - Sin^2(A)

Substitute the value of Sin(A) we found earlier:

Cos^2(A) = 1 - [(5 - 4sqrt(6)) / (sqrt(3) + sqrt(2))]^2

Simplifying this expression will give us the exact value of cos(A).