1. Use each trig ratio to determine all values of theta, to the nearest degree for
0<=theta<=360.
b)tan(theta)=2.1(calc)
To find all values of theta for which tan(theta) is approximately equal to 2.1, we can use the inverse tangent function (arctan) to find the corresponding angles. Since tan(theta) = opposite/adjacent, we can describe the angle as:
θ = arctan(2.1)
Using a calculator, we find that arctan(2.1) is approximately 65.51 degrees.
But we need to find all values of theta in the range 0 <= theta <= 360 degrees, so we need to find the reference angle (angle between 0 and 90 degrees) that corresponds to the given tangent value. Since tangent is positive in both the first and third quadrants, we will have two possible solutions: one in the first quadrant and one in the third quadrant.
First Quadrant Solution:
θ = arctan(2.1) ≈ 65.51 degrees
Third Quadrant Solution:
To find the angle in the third quadrant, we subtract the angle found in the first quadrant from 180 degrees (since the two angles are supplementary):
θ = 180 - arctan(2.1) ≈ 180 - 65.51 ≈ 114.49 degrees
So, the values of theta (to the nearest degree) for tan(theta) ≈ 2.1 in the range 0 <= theta <= 360 are approximately 66 degrees and 114 degrees.