If cos(A+B)=0 and sin(A−B)=21, then find the value of A and B, where A and B are acute angles.
(trigonometry, acute angles, sum and difference identities, sine, cosine)
Get the complete, step-by-step math solution for: "If cos(A+B) = 0 and sin(A-B) = (1)/(2), then find the value of A and B, where A and B are acute angles. (trigonometry, acute angles, sum and differenc...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
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∘ (or π/2 radians). Since A and B are acute angles, their sum A+B must be between 0∘ and 180∘. Therefore, A+B=90∘. This gives us our first equation.
Step 2: Determine A-B from sine equation
We are given that sin(A−B)=21. We know that the sine function is 21 at 30∘ (or π/6 radians). Since A and B are acute angles, A-B must be between −90∘ and 90∘. Therefore, A−B=30∘. 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∘ and A−B=30∘. We can solve this system by adding the two equations together. Adding (A+B) and (A-B) gives 2A, and adding 90∘ and 30∘ gives 120∘. So, 2A=120∘, which means A=60∘.
Step 4: Substitute A to find B
Substitute the value of A=60∘ into the first equation, A+B=90∘. This gives 60∘+B=90∘. Subtracting 60∘ from both sides, we find B=30∘. Both A=60∘ and B=30∘ are acute angles, satisfying the problem conditions.