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Difficulty: MediumTrigonometric Graphs and Simple Equations

What are all the values of θ\theta in the interval 0∘≤θ≤360∘0^\circ \le \theta \le 360^\circ that satisfy the trigonometric equation 4sin⁡2θ−3=04\sin^2\theta - 3 = 0?

  1. 60∘,120∘,240∘,300∘60^\circ, 120^\circ, 240^\circ, 300^\circAnswer
  2. B
    60∘ and 120∘60^\circ \text{ and } 120^\circ
  3. C
    30∘,150∘,210∘,330∘30^\circ, 150^\circ, 210^\circ, 330^\circ
  4. D
    60∘ and 300∘60^\circ \text{ and } 300^\circ

Answer

60∘,120∘,240∘,300∘60^\circ, 120^\circ, 240^\circ, 300^\circ
Solving 4sin⁡2θ−3=04\sin^2\theta - 3 = 0 gives sin⁡2θ=34\sin^2\theta = \frac{3}{4}, so sin⁡θ=±32\sin\theta = \pm\frac{\sqrt{3}}{2}. The reference angle for which sin⁡θ=32\sin\theta = \frac{\sqrt{3}}{2} is 60∘60^\circ. The positive root sin⁡θ=+32\sin\theta = +\frac{\sqrt{3}}{2} gives solutions in Quadrants I and II: 60∘60^\circ and 180∘−60∘=120∘180^\circ - 60^\circ = 120^\circ. The negative root sin⁡θ=−32\sin\theta = -\frac{\sqrt{3}}{2} gives solutions in Quadrants III and IV: 180∘+60∘=240∘180^\circ + 60^\circ = 240^\circ and 360∘−60∘=300∘360^\circ - 60^\circ = 300^\circ. Combining these yields all four angles: 60∘,120∘,240∘,300∘60^\circ, 120^\circ, 240^\circ, 300^\circ.

Step-by-Step Solution

1
Isolate the squared trigonometric term in the equation
4sin⁡2θ=3  ⟹  sin⁡2θ=344\sin^2\theta = 3 \implies \sin^2\theta = \frac{3}{4}
Rearranging the equation allows solving for sin⁡θ\sin\theta directly.
2
Take the square root of both sides, keeping both positive and negative roots
sin⁡θ=±34=±32\sin\theta = \pm\sqrt{\frac{3}{4}} = \pm\frac{\sqrt{3}}{2}
Taking the square root of a squared quantity yields both positive and negative values.
3
Find solutions for sin⁡θ=+32\sin\theta = +\frac{\sqrt{3}}{2} in Quadrants I and II
\theta = 60^\circ \text{ and } \theta = 180^\circ - 60^\circ = 120^\circ
Sine is positive in Quadrants I and II.
4
Find solutions for sin⁡θ=−32\sin\theta = -\frac{\sqrt{3}}{2} in Quadrants III and IV
\theta = 180^\circ + 60^\circ = 240^\circ \text{ and } \theta = 360^\circ - 60^\circ = 300^\circ
Sine is negative in Quadrants III and IV.
5
Combine all solutions within the domain 0∘≤θ≤360∘0^\circ \le \theta \le 360^\circ
\theta \in \{60^\circ, 120^\circ, 240^\circ, 300^\circ\}
All four angles satisfy the original quadratic trigonometric equation.

Key Concept

Solving quadratic trigonometric equations by finding reference angles and evaluating solutions across all four quadrants
Estimated Time:1m 30s
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Topics

Trigonometric Graphs and Simple Equations

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