Find all angles, 0, degrees, is less than or equal to, A, is less than, 360, degrees0 ∘ ≤A<360 ∘ , that satisfy the equation below, to the nearest tenth of a degree (if necessary).2, tangent, A, plus, 14, equals, 5, tangent, A, plus, 82tanA+14=5tanA+8
Question
Find all angles, 0, degrees, is less than or equal to, A, is less than, 360, degrees0 ∘ ≤A<360 ∘ , that satisfy the equation below, to the nearest tenth of a degree (if necessary).2, tangent, A, plus, 14, equals, 5, tangent, A, plus, 82tanA+14=5tanA+8
Solution
To solve the equation 2tanA + 14 = 5tanA + 8, follow these steps:
Step 1: Rearrange the equation to get all terms involving A on one side. This gives us:
5tanA - 2tanA = 14 - 8
Step 2: Simplify the equation:
3tanA = 6
Step 3: Solve for tanA:
tanA = 6 / 3 = 2
Step 4: Now we need to find all angles A that have a tangent of 2. The primary solution is found using the arctan function:
A = arctan(2)
This gives us a primary solution of A = 63.4 degrees (to the nearest tenth of a degree).
Step 5: However, the tangent function has a period of 180 degrees, so we need to add 180 degrees to our primary solution to find additional solutions within the specified range of 0 degrees ≤ A < 360 degrees:
A = 63.4 degrees + 180 degrees = 243.4 degrees
So, the solutions to the equation 2tanA + 14 = 5tanA + 8 within the range 0 degrees ≤ A < 360 degrees are A = 63.4 degrees and A = 243.4 degrees.
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