Solving Trigonometric Equations to Find Angles

In this article, we will solve a set of trigonometric equations to find the values of the angles (A), (B), and (C), given that (sin(BC - A) cos(CA - B) tan(AB - C) 1). This problem involves a series of algebraic and trigonometric manipulations to find the values of the angles.

Understanding the Equations

The given equations are:

(sin(BC - A) 1) (cos(CA - B) 1) (tan(AB - C) 1)

These equations imply specific values for the arguments of the trigonometric functions, based on the standard values where these functions equal 1.

Solving the Equations

We will start by solving each equation separately:

(sin(BC - A) 1) (cos(CA - B) 1) (tan(AB - C) 1)

Solving (sin(BC - A) 1)

We know that (sin(90^circ) 1). Therefore, (BC - A 90^circ). Rearranging the equation, we get (B C - A 90^circ).

Solving (cos(CA - B) 1)

We know that (cos(0^circ) 1). Therefore, (CA - B 0^circ). Rearranging the equation, we get (C A - B 0^circ), or (C B - A).

Solving (tan(AB - C) 1)

We know that (tan(45^circ) 1). Therefore, (AB - C 45^circ). Rearranging the equation, we get (A B - C 45^circ).

Combining the Equations

Now, we have three equations:

(B C - A 90^circ) (C B - A) (A B - C 45^circ)

Step 1: Substitute and Simplify

From equation 2, (C B - A). Substitute this into the first equation:

[begin{align*} B (B - A) - A 90^circ 2B - 2A 90^circ B - A 45^circend{align*}]

From this, we can express (B) in terms of (A):

[B A 45^circ]

Step 2: Substitute (B) into the Second Equation

Substituting (B A 45^circ) into the expression for (C):

[C (A 45^circ) - A 45^circ]

Step 3: Substitute (C) into the Third Equation

Now substitute (C 45^circ) into the third equation:

[A (A 45^circ) - 45^circ 45^circ][2A 45^circ][A 22.5^circ]

Step 4: Find (B)

Now we can find (B) from (B A 45^circ):

[B 22.5^circ 45^circ 67.5^circ]

Step 5: Verify (C)

We already found (C 45^circ).

Final Values of (A), (B), and (C)

The final values of the angles are:

(A 22.5^circ) (B 67.5^circ) (C 45^circ)

These angles are all positive acute angles.

Conclusion

We have verified that the angles (A 22.5^circ), (B 67.5^circ), and (C 45^circ) satisfy the given trigonometric equations. Thus, the solution is:

[boxed{22.5^circ 67.5^circ 45^circ}]