Find all angles in degrees that satisfy the equation. Round approximate answers to the nearest tenth of a degree.
tan α = -1.85
arctan(1.85) = 61.6°
tan(x) < 0 in QII, QIV, so
x = (180-61.6) = 118.4°
x = (360-61.6) = 298.4°
To find all the angles in degrees that satisfy the equation tan α = -1.85, we need to use the inverse tangent function (also known as arctan or tan^(-1)).
The inverse tangent function allows us to find the angle associated with a given tangent value. However, note that the tangent function is periodic, meaning it repeats its values every 180 degrees (or π radians).
To find the angles, we can use the following steps:
1. Using a calculator or a trigonometric table, find the inverse tangent value (arctan) of -1.85. Let's call this value α_0.
2. Since the tangent function is periodic, we need to find the general solutions for α. To do this, we add or subtract a multiple of 180 degrees (or π radians) to the initial solution α_0.
3. To obtain all the angles that satisfy the equation, continue adding or subtracting 180 degrees (or π radians) to α_0 as necessary.
4. Round each angle to the nearest tenth of a degree to match the desired level of precision.
Let's perform the calculations:
Using a calculator or a trigonometric table, the approximate value of the inverse tangent of -1.85 is -61.43 degrees.
Now, let's find the general solutions by adding or subtracting multiples of 180 degrees:
- α = -61.43° + 180° = 118.57°
- α = -61.43° + 2 * 180° = 298.57°
- α = -61.43° + 3 * 180° = 478.57°
- ...
Continue this pattern to find all possible solutions. Then, round each angle to the nearest tenth of a degree.
Therefore, the angles that satisfy the equation tan α = -1.85, rounded to the nearest tenth of a degree, are:
- α ≈ -61.4°
- α ≈ 118.6°
- α ≈ 298.6°
- α ≈ 478.6°
- ...