Find all angles, 0≤A<360, that satisfy the equation below, to the nearest 10th of a degree.
2tanA+6=tanA+3
2tanA+6=tanA+3
tanA = -3
so A must be in II or IV
using tanx = +3
x = 71.565°
so A = 180-71.564 = appr 108.4
or
A = 360 - 71.565 = appr 288.4°
To find all the angles A that satisfy the equation 2tan(A) + 6 = tan(A) + 3, we need to solve for A. Here are the steps:
Step 1: Simplify the equation:
2tan(A) + 6 = tan(A) + 3
Subtract tan(A) from both sides:
2tan(A) - tan(A) + 6 = 3
Simplify:
tan(A) + 6 = 3
Step 2: Subtract 6 from both sides:
tan(A) + 6 - 6 = 3 - 6
Simplify:
tan(A) = -3
Step 3: Take the inverse tangent of both sides to find A:
A = atan(-3) ≈ -71.57°
Step 4: Since we are looking for all angles between 0 and 360 degrees, we need to add or subtract multiples of 180 degrees to the solution from step 3:
A1 = -71.57° + 180° = 108.43°
A2 = -71.57° + 360° = 288.43°
So, the angles that satisfy the equation are approximately 108.4° and 288.4° to the nearest tenth of a degree.
To solve the equation 2tanA + 6 = tanA + 3, we need to isolate the variable A.
Step 1: Subtract tanA from both sides:
2tanA - tanA + 6 = 3
This simplifies to:
tanA + 6 = 3
Step 2: Subtract 6 from both sides:
tanA + 6 - 6 = 3 - 6
This simplifies to:
tanA = -3
Now, we need to find the values of A that satisfy the equation in the given interval of 0 ≤ A < 360.
Step 3: Calculate the angle A using the inverse tangent function (tan⁻¹) on both sides:
A = tan⁻¹(-3)
Using a calculator, find the principal value of the inverse tangent of -3, which is approximately -71.57 degrees.
Note: The inverse tangent function gives an angle in the range of -90° to 90° (-π/2 to π/2 in radians). However, we need angles in the range of 0° to 360° (0 to 2π in radians).
Step 4: To find other solutions within the given interval, add or subtract multiples of 180° (π radians) to the principal value.
Adding 180° to -71.57°, we get:
-71.57° + 180° ≈ 108.43°
So, one solution is A ≈ 108.4°.
Adding another 180° to 108.43°, we get:
108.43° + 180° = 288.43°
Therefore, another solution is A ≈ 288.4°.
Hence, the angles that satisfy the equation 2tanA + 6 = tanA + 3, to the nearest 10th of a degree, are approximately 108.4° and 288.4°.