Part 1Find thetaθ to four significant digits for 0less than or equals≤thetaθless than

Question

Part 1Find thetaθ to four significant digits for 0less than or equals≤thetaθless than<2piπ if cosine thetacosθequals=0.71990.7199. Part 1thetaθequals=[input]enter your response here ​(Simplify your answer. Use a comma to separate answers as​ needed.)

BlackTom AI · Accepted Answer

Cosine of θ equals 0.7199, so the principal angle is θ = arccos(0.7199).

1. Relevant knowledge points and concepts
- The arccos (inverse cosine) function gives the principal angle in [0, π] whose cosine equals the given value.
- Cosine is even and 2π-periodic: cos(θ) = cos(−θ) and cos(θ) = cos(2π − θ). For a given value y with −1 ≤ y ≤ 1, within [0, 2π) there are at most two solutions: θ = arccos(y) and θ = 2π − arccos(y).

2. Step-by-step reasoning
- Compute the principal solution θ1 = arccos(0.7199).
- Compute the second solution in [0, 2π) using θ2 = 2π − θ1.

3. Calculations and intermediate steps
- Using a calculator: θ1 = arccos(0.7199) ≈ 0.767094... radians.
- Round θ1 to four significant digits: 0.7671.
- 2π = 6.283185307...
- θ2 = 2π − θ1 ≈ 6.283185307 − 0.767094... = 5.5160912...
- Round θ2 to four significant digits: 5.516.

4. Why the correct answers are correct
- θ1 = 0.7671 is the principal arccos value, and cos(0.7671) ≈ 0.7199 (to the given precision).
- θ2 = 5.516 is the symmetric angle in [0, 2π) that has the same cosine because cos(2π − θ1) = cos θ1.

5. Why other potential angles are incorrect
- There are no other distinct solutions in [0, 2π) because cosine repeats every 2π and is symmetric about the x-axis; the only values in one period that yield a given cosine value are θ1 and 2π − θ1.
- Angles like π ± θ1 would produce cosines of −cos(θ1), not +cos(θ1), so they are not solutions.

6. Conclusion / final answer
- The two solutions in 0 ≤ θ < 2π, rounded to four significant digits, are 0.7671 and 5.516, which matches the provided answer.

Part 1Find thetaθ to four significant digits for 0less than or equals≤thetaθless than<2piπ if cosine thetacosθequals=0.71990.7199. Part 1thetaθequals=[input]enter your response here (Simplify your answer. Use a comma to separate answers as needed.)多项填空题

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Part 1Find thetaθ to four significant digits for 0less than or equals≤thetaθless than<2piπ if cosine thetacosθequals=0.71990.7199. Part 1thetaθequals=[input]enter your response here ​(Simplify your answer. Use a comma to separate answers as​ needed.)多项填空题