Find the exact value of
Sin ( sin-1. 9/41 - cos-1 (-5/13)
let sin-1. 9/41 = ß and cos-1 (-5/13 = Ø
so sin ß = 9/41 and cos ß = 40/41
and cosØ = -5/13 and sinØ = 12/13
then Sin ( sin-1. 9/41 - cos-1 (-5/13)
= sin(ß - Ø)
= sinßcosØ - cosßsinØ
= (9/41)(-5/13) - (40/41)(12/13) = -525/533
To find the exact value of sin(sin^⁻1(9/41) - cos^⁻1(-5/13)), we can use the Pythagorean identity and the angle addition formula.
First, let's find the values of sin^⁻1(9/41) and cos^⁻1(-5/13).
The value of sin^⁻1(x) represents the angle whose sine is x. Similarly, cos^⁻1(x) represents the angle whose cosine is x.
Using a calculator or trigonometric table, we can find:
sin^⁻1(9/41) ≈ 13.333 degrees
cos^⁻1(-5/13) ≈ 116.565 degrees
Now, let's use these values to determine the angle for which we are calculating the sine:
θ = sin^⁻1(9/41) - cos^⁻1(-5/13)
θ ≈ 13.333 - 116.565
θ ≈ -103.232 degrees
Since sine is an odd function, we can find the exact value of sin(θ) by looking at the corresponding angle in the unit circle. In this case, sin(θ) = sin(-103.232).
In the fourth quadrant of the unit circle, the sine function is negative. Thus, we can write:
sin(θ) = -sin(103.232)
Now, using a calculator or trigonometric table:
sin(103.232) ≈ 0.901
Therefore, the exact value of sin(sin^⁻1(9/41) - cos^⁻1(-5/13)) is approximately -0.901.
To find the exact value of the expression Sin ( sin^(-1)(9/41) - cos^(-1)(-5/13)), we can use the following steps:
Step 1: Use the identity sin^(-1)(x) = arcsin(x) to convert sin^(-1)(9/41) to arcsin(9/41).
Step 2: Use the identity cos^(-1)(x) = arccos(x) to convert cos^(-1)(-5/13) to arccos(-5/13).
Step 3: Substitute the converted values into the original expression: Sin ( arcsin(9/41) - arccos(-5/13)).
Step 4: Apply the angle subtraction identity for sine function, which states that Sin(a - b) = Sin(a) * Cos(b) - Cos(a) * Sin(b).
Applying this identity, the expression becomes:
Sin ( arcsin(9/41) - arccos(-5/13)) = Sin (arcsin(9/41)) * Cos(arccos(-5/13)) - Cos(arcsin(9/41)) * Sin(arccos(-5/13)).
Step 5: Now, we can evaluate the individual trigonometric functions using the given angles:
- For arcsin(9/41), since arcsin gives the angle whose sine is 9/41, we have Sin(arcsin(9/41)) = 9/41.
- For arccos(-5/13), since arccos gives the angle whose cosine is -5/13, we have Cos(arccos(-5/13)) = -5/13.
Substituting these values back into the expression, we have:
Sin ( arcsin(9/41) - arccos(-5/13)) = (9/41) * (-5/13) - Cos(arcsin(9/41)) * Sin(arccos(-5/13)).
Step 6: To find the value of Cos(arcsin(9/41)) * Sin(arccos(-5/13)), we can use the Pythagorean identity for sine and cosine. The identity states that Cos^2(x) + Sin^2(x) = 1.
So, we have Cos(arcsin(9/41)) = sqrt(1 - (9/41)^2) and Sin(arccos(-5/13)) = sqrt(1 - (-5/13)^2).
Step 7: Evaluate the square roots and substitute the values back into the expression:
Sin( arcsin(9/41) - arccos(-5/13)) = (9/41) * (-5/13) - sqrt(1 - (9/41)^2) * sqrt(1 - (-5/13)^2).
Step 8: Simplify and compute the values:
Sin( arcsin(9/41) - arccos(-5/13)) = (9/41) * (-5/13) - sqrt(1 - (9/41)^2) * sqrt(1 - (-5/13)^2).
= -45/533 - sqrt(1 - 81/1681) * sqrt(1 - 25/169).
= -45/533 - sqrt(1600/1681) * sqrt(1444/1681).
= -45/533 - 40/41 * 38/41.
= -45/533 - 1520/1681.
= -6675/79003 ≈ -0.0844 (rounded to four decimal places).
Therefore, the exact value of Sin ( sin^(-1)(9/41) - cos^(-1)(-5/13)) is approximately -0.0844.