(a) If cos(A+B)=21 and tan(A−B)=31, where 0≤A+B≤90∘, then find the value of sec(2A−3B).
OR
(b) Find the value of x such that, 3tan260∘−xsin245∘+34sec230∘=2 cosec230∘
Get the complete, step-by-step math solution for: "(a) If cos (A + B) = (1)/(2) and tan (A - B) = {1}{√(3)}, where 0 ≤ A + B ≤ 90^ , then find the value of sec (2A - 3B). OR (b) Find the value of x suc...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Determine A + B and A - B
Given the trigonometric equations, we can find the values of A+B and A-B. Since cos60∘=21 and tan30∘=31, we deduce that A+B=60∘ and A−B=30∘.
Step 2: Solve for A and B
We have a system of two linear equations: A+B=60∘ and A−B=30∘. Adding these two equations eliminates B, giving 2A=90∘, so A=45∘. Substituting A=45∘ into the first equation yields 45∘+B=60∘, which means B=15∘.
Step 3: Calculate 2A - 3B
Now that we have the values of A and B, we can calculate the expression 2A - 3B. Substitute A=45∘ and B=15∘ into the expression to get 2(45∘)−3(15∘), which simplifies to 90∘−45∘=45∘.
Step 4: Find the value of sec(2A - 3B)
Finally, we need to find the value of sec(2A−3B). Since 2A−3B=45∘, we need to calculate sec45∘. We know that secθ=cosθ1, and cos45∘=21. Therefore, sec45∘=2.