What are all values of θ, −π ≤ θ ≤ π, for which 2 cos θ > -1 and 2 sin θ > √3?
Understand the Problem
The question is asking for all values of theta within the specified range, under certain trigonometric conditions involving cosine and sine functions. We will need to analyze the inequalities given and solve for theta accordingly.
Answer
$$ \frac{\pi}{3} < \theta < \frac{2\pi}{3} $$
Answer for screen readers
The values of $\theta$ that satisfy both inequalities are: $$ \frac{\pi}{3} < \theta < \frac{2\pi}{3} $$
Steps to Solve
- Analyze the first inequality: $2 \cos \theta > -1$
To solve $2 \cos \theta > -1$, divide both sides by 2: $$ \cos \theta > -\frac{1}{2} $$
The cosine function is greater than $-\frac{1}{2}$ in the intervals: $$ \theta \in \left(-\frac{5\pi}{6}, \frac{5\pi}{6}\right) $$
- Analyze the second inequality: $2 \sin \theta > \sqrt{3}$
Next, solve $2 \sin \theta > \sqrt{3}$ by dividing both sides by 2: $$ \sin \theta > \frac{\sqrt{3}}{2} $$
The sine function is greater than $\frac{\sqrt{3}}{2}$ in the intervals: $$ \theta \in \left(\frac{\pi}{3}, \frac{2\pi}{3}\right) $$
- Combine the two results
Now, we need to find the intersection of the two intervals:
- From the first inequality: $\theta \in \left(-\frac{5\pi}{6}, \frac{5\pi}{6}\right)$
- From the second inequality: $\theta \in \left(\frac{\pi}{3}, \frac{2\pi}{3}\right)$
The overlapping portion of these intervals is: $$ \theta \in \left(\frac{\pi}{3}, \frac{2\pi}{3}\right) $$
The values of $\theta$ that satisfy both inequalities are: $$ \frac{\pi}{3} < \theta < \frac{2\pi}{3} $$
More Information
The inequalities indicate the ranges where the functions $\cos \theta$ and $\sin \theta$ meet the specified criteria. The values describe where these trigonometric functions yield values within the specified limits.
Tips
- Ignoring interval bounds: When combining intervals, it’s essential to find the intersection accurately.
- Misapplying trigonometric values: Make sure to remember the unit circle to reference where sine and cosine are positive or negative.
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