Find all solutions of the equation in the interval [0, 2π]. cosθ = -0.4357. If there is more than one solution, separate them with commas.
Understand the Problem
The question is asking to find all angles theta (θ) in the specified interval [0, 2π] where the cosine of theta equals -0.4357. This involves using trigonometric principles to determine the angles that satisfy this equation.
Answer
The solutions are approximately $3.012, 4.270$ radians.
Answer for screen readers
The solutions are approximately $3.012, 4.270$ radians.
Steps to Solve
- Determine the reference angle
To find the angles where the cosine is negative, we first find the reference angle by taking the inverse cosine (arccos) of the absolute value of -0.4357:
$$ \theta_{\text{ref}} = \cos^{-1}(0.4357) $$
- Calculate the reference angle
Using a calculator, we find:
$$ \theta_{\text{ref}} \approx 1.129 \text{ radians} $$
- Locate the angles in the unit circle
In the interval $[0, 2\pi]$, cosine is negative in the second and third quadrants. Therefore, we can find the corresponding angles as:
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Second quadrant: $$ \theta_1 = \pi - \theta_{\text{ref}} $$
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Third quadrant: $$ \theta_2 = \pi + \theta_{\text{ref}} $$
Substituting the reference angle back into these equations gives:
$$ \theta_1 = \pi - 1.129 $$ $$ \theta_2 = \pi + 1.129 $$
- Calculate the specific angles
Now we evaluate these expressions:
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For $\theta_1$: $$ \theta_1 \approx \pi - 1.129 \approx 3.012 \text{ radians} $$
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For $\theta_2$: $$ \theta_2 \approx \pi + 1.129 \approx 4.270 \text{ radians} $$
The solutions are approximately $3.012, 4.270$ radians.
More Information
The cosine function is periodic and takes on both positive and negative values depending on the quadrant. The angles found correspond to the locations on the unit circle where the cosine values are negative.
Tips
- Failing to consider both quadrants where cosine is negative.
- Miscalculating the reference angle using the cosine inverse.
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