Question textDetermine all solutions of the following equation for ..Input your solutions, from smallest to largest, in the boxes below. Remember you can input by typing 'pi'. Thus the expression would be inputted as '3pi/2'. Answer 1 Question 13[input] or Answer 2 Question 13[input]

Question

BlackTom AI · Accepted Answer

The sine function equals −1 only at the angle 3π/2 (plus full rotations). So set the argument equal to that principal value plus 2kπ:
1) Relevant concepts
- sin x = −1 exactly when x = 3π/2 + 2kπ (k ∈ Z).
- Solve 2θ + π/4 = 3π/2 + 2kπ and restrict θ to [0,2π].

2) Step-by-step algebra and intermediate steps
2θ + π/4 = 3π/2 + 2kπ
Subtract π/4 from both sides:
2θ = 3π/2 − π/4 + 2kπ
Find a common denominator (4):
3π/2 = 6π/4, so 6π/4 − π/4 = 5π/4
Thus 2θ = 5π/4 + 2kπ
Divide by 2:
θ = (5π/4)/2 + kπ = 5π/8 + kπ

3) Find values in [0,2π]
Take integer k values:
- k = 0: θ = 5π/8 (≈ 0.625π, within [0,2π])
- k = 1: θ = 5π/8 + π = 5π/8 + 8π/8 = 13π/8 (≈ 1.625π, within [0,2π])
- k = −1: θ = 5π/8 − π = −3π/8 (negative, outside interval)
- k = 2: θ = 5π/8 + 2π = 21π/8 (> 2π, outside interval)
So the only solutions in [0,2π] are θ = 5π/8 and θ = 13π/8.

4) Verification (why these are correct)
Check θ = 5π/8: 2θ + π/4 = 10π/8 + 2π/8 = 12π/8 = 3π/2 → sin(3π/2) = −1.
Check θ = 13π/8: 2θ + π/4 = 26π/8 + 2π/8 = 28π/8 = 7π/2 = 3π/2 + 2π → sin = −1.

5) Why other possibilities are excluded
No other integer k yields θ inside [0,2π]; choices like k = −1 and k = 2 give values outside the interval, so they are not valid solutions here.

Conclusion: The solutions in increasing order are 5pi/8 and 13pi/8, matching the provided answer.

Question textDetermine all solutions of the following equation for ..Input your solutions, from smallest to largest, in the boxes below. Remember you can input by typing 'pi'. Thus the expression would be inputted as '3pi/2'. Answer 1 Question 13[input] or Answer 2 Question 13[input]多项填空题

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