Find all solutions of the equation in the interval [0, 2π]. secθ = 1.4671. If there is more than one solution, separate them with commas.
Understand the Problem
The question is asking to find all angles θ within the interval [0, 2π] that satisfy the equation sec(θ) = 1.4671. To solve this, we will use the relationship between secant and cosine, as well as the unit circle to find the appropriate values of θ.
Answer
$0.8148, 5.4683$
Answer for screen readers
The solutions are $0.8148, 5.4683$.
Steps to Solve
- Understanding secant and cosine relation
The secant function is the reciprocal of the cosine function. Thus, we can rewrite the equation as:
$$ \sec(\theta) = 1.4671 \implies \cos(\theta) = \frac{1}{1.4671} $$
- Calculating cosine value
Now, we need to find the cosine value:
$$ \cos(\theta) = \frac{1}{1.4671} \approx 0.6828 $$
- Finding angles from cosine values
We will use the inverse cosine function to find the principal value of $\theta$:
$$ \theta = \cos^{-1}(0.6828) $$
Calculating this gives:
$$ \theta_1 \approx 0.8148 \quad \text{(in the first quadrant)} $$
- Determining the second angle
Since cosine is positive in both the first and fourth quadrants, we find the second solution using symmetry:
$$ \theta_2 = 2\pi - \theta_1 \approx 2\pi - 0.8148 \approx 5.4683 $$
- Listing all solutions
Thus, the solutions for the original equation $\sec(\theta) = 1.4671$ in the interval $[0, 2\pi]$ are:
$$ \theta \approx 0.8148, 5.4683 $$
The solutions are $0.8148, 5.4683$.
More Information
This problem illustrates the relationship between the secant and cosine functions, as well as how to find angles using the unit circle. The calculations demonstrate that there can be two valid angles for a given cosine value, given the periodic nature of trigonometric functions.
Tips
- Confusing the quadrants: Remember that cosine is positive in the first and fourth quadrants.
- Miscalculating the cosine value leading to wrong angle results.
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