sin [sin -1 (7/25) - cos -1 (-8/17)]

let A = sin^-1 (7/25)

then cosA = 24/25 ---> sinA = 7/25

let B = cos^-1 (-8/17)
cosB = --8/17 , let's assume B is in II
sin B = 15/17

so sin [sin -1 (7/25) - cos -1 (-8/17)]
= sin (A - B)
= sinAcosB - cosAsinB
= (7/25)(-8/17) - (24/25)(15/17)
= (-56 - 360)/425
= -416/425

check with calculator:
A = 16.26°
B = 118.07°
A-B = -101.81°
sin(-101.81°) = -.9788
-416/425 = -.9788 , how about that?

To solve the expression sin[sin^(-1)(7/25) - cos^(-1)(-8/17)], we can follow these steps:

Step 1: We know that sin^-1(x) represents the inverse sine function, which returns the angle whose sine is x. Similarly, cos^-1(x) represents the inverse cosine function, which returns the angle whose cosine is x.

Step 2: Let's calculate the inner part: sin^-1(7/25) - cos^-1(-8/17).

Step 3: Using a calculator or reference table, we can find that sin^-1(7/25) is approximately 0.2827 radians or 16.18 degrees.

Step 4: Similarly, cos^-1(-8/17) is approximately 2.3317 radians or 133.81 degrees.

Step 5: Now we substitute these values back into the original expression: sin(0.2827 - 2.3317).

Step 6: Subtracting the values inside the sin function gives us approximately -2.049 radians or -117.63 degrees.

Therefore, sin[sin^(-1)(7/25) - cos^(-1)(-8/17)] is approximately -2.049 radians or -117.63 degrees.

To find the value of the expression sin[sin^(-1)(7/25) - cos^(-1)(-8/17)], we can use the properties of trigonometric functions.

Let's break it down step by step:

Step 1: Evaluate sin^(-1)(7/25)
The notation sin^(-1)(x) represents the inverse sine function, or arcsin. It gives us the angle whose sine is x.
So, sin^(-1)(7/25) means finding the angle whose sine is 7/25.

To do this, you need to use a calculator or a trigonometric table that has inverse functions like arcsin. Plug in 7/25 into your calculator or trigonometric table and calculate the inverse sine value, which will give you an angle.

Let's say the inverse sine of 7/25 is A radians (A° in degrees).

Step 2: Evaluate cos^(-1)(-8/17)
Again, cos^(-1)(x) represents the inverse cosine function, or arccos. It gives us the angle whose cosine is x.
So, cos^(-1)(-8/17) means finding the angle whose cosine is -8/17.

Similar to step 1, use a calculator or trigonometric table to calculate the inverse cosine of -8/17, which will give you another angle.

Let's say the inverse cosine of -8/17 is B radians (B° in degrees).

Step 3: Calculate the expression sin(A - B)
Now that we have A and B (angles in radians), we can calculate the expression sin(A - B).

Use the property of the sine function: sin(A - B) = sin(A) * cos(B) - cos(A) * sin(B)

Plug in the values of A and B that you found in steps 1 and 2, respectively, and evaluate this expression.

This will give you the final result for the given expression sin[sin^(-1)(7/25) - cos^(-1)(-8/17)].