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Find all angles, 0^{\circ} \leq \theta<360^{\circ} , that satisfy the equation below, to the nearest tenth of a degree.Answer:
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Q. Find all angles, , that satisfy the equation below, to the nearest tenth of a degree.Answer:
- Solve Quadratic Equation: Let's first solve the quadratic equation in terms of . The equation is . This is a quadratic equation in standard form, where , , and .
- Factor Quadratic Equation: We can factor the quadratic equation as . This gives us two possible solutions for : and .
- Find Theta for Cot(): Now we need to find the angles that correspond to these cotangent values. For , we know that cotangent is the reciprocal of tangent, so we are looking for angles where . The angles with tangent of in the range of to degrees are degrees and degrees.
- Find Theta for Cot(): For , we need to use a calculator to find the angles that correspond to this cotangent value. Since cotangent is the reciprocal of tangent, we are looking for angles where . We can use the inverse tangent function to find the reference angle, and then determine the angles in the specified range that have this tangent value.
- Combine Results: Using a calculator, we find that the reference angle for is approximately degrees. Since the tangent is negative in the second and fourth quadrants, the angles that satisfy this condition are degrees and degrees.
- Combine Results: Using a calculator, we find that the reference angle for is approximately degrees. Since the tangent is negative in the second and fourth quadrants, the angles that satisfy this condition are degrees and degrees.Combining the results from the two factors, we have four angles that satisfy the original equation: degrees, degrees, degrees, and degrees.
