Solving Trigonometric Equations to Find Angle A

This problem involves solving a system of trigonometric equations to find the values of angles A and B, given specific conditions. We are given two equations relating trigonometric functions of sums and differences of angles A and B, along with constraints on the angles themselves.

Understanding the Given Information

• Equation 1: $\backslash sin\; \backslash left(\; \{A\; -\; B\}\; \backslash right)\; =\; \backslash frac\{1\}\{2\}$
• Equation 2: $\backslash cos\; \backslash left(\; \{A\; +\; B\}\; \backslash right)\; =\; \backslash frac\{1\}\{2\}$
• Condition 1: $A\; >\; B\; >\; 0^\backslash circ$
• Condition 2: $A\; +\; B$ is an acute angle (meaning )

Finding the Angles A - B and A + B

We need to find the angles whose sine and cosine values match the given fractions. We use our knowledge of standard trigonometric angles.

From Equation 1: $\backslash sin\; \backslash left(\; \{A\; -\; B\}\; \backslash right)\; =\; \backslash frac\{1\}\{2\}$

We know that $\backslash sin\; \backslash left(\; \{30^\backslash circ\}\; \backslash right)\; =\; \backslash frac\{1\}\{2\}$. Therefore, one possible value for $A\; -\; B$ is $30^\backslash circ$. The condition $A\; >\; B$ implies $A\; -\; B\; >\; 0$, so $30^\backslash circ$ is a valid positive difference. Also, since $A+B$ is acute, and (assuming B > 0, which is given), $A-B$ must also be less than $90^\backslash circ$. Thus, $A\; -\; B\; =\; 30^\backslash circ$ is the value that satisfies the conditions.

So, we have: $$A\; -\; B\; =\; 30^\backslash circ\; \backslash quad\; \backslash text\{(Equation\; 3)\}$$

From Equation 2: $\backslash cos\; \backslash left(\; \{A\; +\; B\}\; \backslash right)\; =\; \backslash frac\{1\}\{2\}$

We know that $\backslash cos\; \backslash left(\; \{60^\backslash circ\}\; \backslash right)\; =\; \backslash frac\{1\}\{2\}$. The condition is that $A\; +\; B$ is an acute angle, which means . The value $60^\backslash circ$ falls within this range and is the angle whose cosine is $\backslash frac\{1\}\{2\}$ in the first quadrant.

So, we have: $$A\; +\; B\; =\; 60^\backslash circ\; \backslash quad\; \backslash text\{(Equation\; 4)\}$$

Solving for A and B

Now we have a system of two linear equations with variables A and B:

• Equation 3: $A\; -\; B\; =\; 30^\backslash circ$
• Equation 4: $A\; +\; B\; =\; 60^\backslash circ$

We can solve this system by adding the two equations:

To find B, substitute the value of A ($45^\backslash circ$) into either Equation 3 or Equation 4. Using Equation 4:

Verification

Let's check if the values $A\; =\; 45^\backslash circ$ and $B\; =\; 15^\backslash circ$ satisfy all the original conditions:

• $\backslash sin(A\; -\; B)\; =\; \backslash sin(45^\backslash circ\; -\; 15^\backslash circ)\; =\; \backslash sin(30^\backslash circ)\; =\; \backslash frac\{1\}\{2\}$ (Matches Equation 1)
• $\backslash cos(A\; +\; B)\; =\; \backslash cos(45^\backslash circ\; +\; 15^\backslash circ)\; =\; \backslash cos(60^\backslash circ)\; =\; \backslash frac\{1\}\{2\}$ (Matches Equation 2)
• $A\; >\; B\; >\; 0^\backslash circ$: $45^\backslash circ\; >\; 15^\backslash circ\; >\; 0^\backslash circ$ (Matches Condition 1)
• $A\; +\; B$ is acute: $A\; +\; B\; =\; 45^\backslash circ\; +\; 15^\backslash circ\; =\; 60^\backslash circ$, and (Matches Condition 2)

All conditions are satisfied. The value of A is $45^\backslash circ$.

Combining the derived relationships:

Adding these equations gives $2A\; =\; 90^\backslash circ$, leading to $A\; =\; 45^\backslash circ$.

Revision Table: Solving Trigonometric Equations

Additional Information: Trigonometric Identities

While not directly used in this specific solution, it's useful to know about trigonometric identities for sums and differences of angles:

• $\backslash sin(x\; +\; y)\; =\; \backslash sin\; x\; \backslash cos\; y\; +\; \backslash cos\; x\; \backslash sin\; y$
• $\backslash sin(x\; -\; y)\; =\; \backslash sin\; x\; \backslash cos\; y\; -\; \backslash cos\; x\; \backslash sin\; y$
• $\backslash cos(x\; +\; y)\; =\; \backslash cos\; x\; \backslash cos\; y\; -\; \backslash sin\; x\; \backslash sin\; y$
• $\backslash cos(x\; -\; y)\; =\; \backslash cos\; x\; \backslash cos\; y\; +\; \backslash sin\; x\; \backslash sin\; y$

These identities can be used in more complex problems involving sums and differences of angles. In this case, recognizing the standard angle values for sine and cosine was sufficient.

Understanding the unit circle also helps visualize the angles where sine and cosine take specific values like $\backslash frac\{1\}\{2\}$.