If $\cos(A+B) = 0$ and $\sin(A-B) = \frac{1}{2}$, then find the value of $A$ and $B$, where $A$ and $B$ are acute angles. (trigonometry, acute angles, sum and difference identities, sine, cosine)

Step 1: Determine A+B from cosine equation

We are given that $\cos(A+B) = 0$. We know that the cosine function is zero at $90^\circ$ (or $\pi/2$ radians). Since $A$ and $B$ are acute angles, their sum $A+B$ must be between $0^\circ$ and $180^\circ$. Therefore, $A+B = 90^\circ$. This gives us our first equation.

Step 2: Determine A-B from sine equation

We are given that $\sin(A-B) = \frac{1}{2}$. We know that the sine function is $\frac{1}{2}$ at $30^\circ$ (or $\pi/6$ radians). Since $A$ and $B$ are acute angles, $A-B$ must be between $-90^\circ$ and $90^\circ$. Therefore, $A-B = 30^\circ$. This is our second equation.

Step 3: Solve the system of linear equations

Now we have a system of two linear equations: $A+B = 90^\circ$ and $A-B = 30^\circ$. We can solve this system by adding the two equations together. Adding $(A+B)$ and $(A-B)$ gives $2A$, and adding $90^\circ$ and $30^\circ$ gives $120^\circ$. So, $2A = 120^\circ$, which means $A = 60^\circ$.

Step 4: Substitute A to find B

Substitute the value of $A = 60^\circ$ into the first equation, $A+B = 90^\circ$. This gives $60^\circ + B = 90^\circ$. Subtracting $60^\circ$ from both sides, we find $B = 30^\circ$. Both $A=60^\circ$ and $B=30^\circ$ are acute angles, satisfying the problem conditions.

More from this chapter

• **EXERCISE 8.3** **1.** Express the trigonometric ratios sin A, tan A and sec A in terms of cot A.
• Cos A / 1 + sin a + 1 + sin a / cos A is equal to 2 sec a
• write all the other trigonometric ratios of angle A in terms of sec A
• Express all the trigonometric ratios in terms of Sin A
• Express the trigonometric ratio sign a sec a tan a in terms of cot a
• Prove that √(sec² A + csc² A) = tan A + cot A
• (secθ - tanθ) = (1)/(m) --- (2) Adding (1) & (2) secθ + tanθ + secθ - tanθ = (1)/(m) + (m)/(1) 2secθ = (1 + m²)/(m) secθ = (1 + m²)/(2m) (Hence proved) 2) If as
• If tanθ + secθ = m, prove that secθ = (m² + 1)/(2m).

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