Answer:
about 37.448° and 217.448°
Step-by-step explanation:
You want the values of θ in the interval [0°, 360°) such that ...
tan(θ) = 0.7658738
The inverse tangent function will give an angle in the range (-90°, 90°). For positive tangent values, the angle will be in the first quadrant. The tangent function is periodic with period 180°, so another angle in the interval of interest will be 180° more than the value returned by the arctangent function.
tan(θ) = 0.7658738
θ = arctan(0.7658738) ≈ 37.448° + n(180°)
θ = {37.448°, 217.448°}
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Additional comment
The second attachment gives the angles to 11 decimal places. Angular measures beyond about 6 decimal places don't have much practical use. My GPS receiver reports my position (latitude, longitude) using 8 decimal places (a resolution of about 0.03 inches), but its error is about 10,000 times that.
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The values of θ that satisfy tan θ = 0.7658738 in the interval [0°, 360°) are approximately: 38.105°, 218.105°, -141.895°. To find the values of θ in the interval [0°, 360°) that satisfy the equation tan θ = 0.7658738, you can use the inverse tangent function (arctan) to find the angle corresponding to the given tangent value.
However, since the tangent function has a periodicity of π (180°), we need to consider all possible angles within the given interval. Let's calculate the inverse tangent of 0.7658738: θ = arctan(0.7658738) ≈ 38.105°.
Now, since the tangent function repeats every 180°, we need to find all other angles that have the same tangent value by adding or subtracting multiples of 180°:
θ = 38.105° + 180° = 218.105°
θ = 38.105° - 180° = -141.895°
In the interval [0°, 360°), the solutions are 38.105°, 218.105°, and their corresponding angles in the negative range, -141.895°. Therefore, the values of θ that satisfy tan θ = 0.7658738 in the interval [0°, 360°) are approximately: 38.105°, 218.105°, -141.895°.
Learn more about inverse tangent function here: brainly.com/question/28540481
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