Question
Find a value of \( s \) in the interval \( \left.\left[0, \frac{\pi}{2}\right] \quad \begin{array}{l}s=\square \\ \text { that satisfies the given statement. } \\ \sec s=1.4782\end{array} \right\rvert\, \begin{array}{l}\text { (Round to four decimal places as needed.) }\end{array} \) |
Ask by Olson Brewer. in the United States
Mar 01,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( s \approx 0.8262 \) radians
Solution
We start with the equation
sec s = 1.4782.
Recall that sec s = 1/cos s, so we can rewrite the equation as
1/cos s = 1.4782 ⟹ cos s = 1/1.4782.
Calculating the right‐hand side gives
cos s ≈ 0.6763.
Now, to find s, take the inverse cosine (arccos) of both sides:
s = arccos(0.6763).
Using a calculator (ensuring it is in radian mode), we get
s ≈ 0.8262 radians.
Rounded to four decimal places, the value of s in the interval [0, π/2] that satisfies sec s = 1.4782 is
s = 0.8262.
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Simplify this solution The Deep Dive
To find the value of \( s \) such that \( \sec s = 1.4782 \), we can use the fact that \( \sec s = \frac{1}{\cos s} \). Therefore, we can rewrite the equation as: \[ \cos s = \frac{1}{1.4782} \] Calculating the value gives us: \[ \cos s \approx 0.6761 \] Now, to find \( s \), we take the inverse cosine: \[ s = \cos^{-1}(0.6761) \] Using a calculator, we find: \[ s \approx 0.8324 \] Thus, the value of \( s \) that satisfies \( \sec s = 1.4782 \) in the interval \( \left[0, \frac{\pi}{2}\right] \) is approximately: \[ \boxed{0.8324} \]
