Leila tried to solve the following equation, where $x$ is in degrees to the nearest tenth, in the interval from $-360^{\circ}$ to $0^{\circ}$, but she made a mistake.
$$
\cos (x)=0.74
$$

Here is Leila's work.
$$
\begin{aligned}
\cos ^{-1}(\cos (x)) & =\cos ^{-1}(0.74) \
x & =42.3 \
x & =42.3+360 n \
x & \approx \ldots-677.7,-317.7,42.3 \ldots
\end{aligned}
$$
$n$ is any integ $\epsilon$

Of the solutions, only -317.7 is in the interval from -360 to 0 , so $x \approx-317.7$.

What was Leila's mistake?
Choose 1 answer:
(A) Leila evaluated $\cos ^{-1}(0.74)$ in radians instead of in degrees.

B Leila forgot to apply an identity which would reveal another solution in the interval.
(c) Leila forgot to use larger $n$-values which would reveal another solution in the interval.

D Leila gave solutions that were outside of the interval.

Question

Leila tried to solve the following equation, where $x$ is in degrees to the nearest tenth, in the interval from $-360^{\circ}$ to $0^{\circ}$, but she made a mistake.
$$
\cos (x)=0.74
$$

Here is Leila's work.
$$
\begin{aligned}
\cos ^{-1}(\cos (x)) & =\cos ^{-1}(0.74) \
x & =42.3 \
x & =42.3+360 n \
x & \approx \ldots-677.7,-317.7,42.3 \ldots
\end{aligned}
$$
$n$ is any integ $\epsilon$

Of the solutions, only -317.7 is in the interval from -360 to 0 , so $x \approx-317.7$.

What was Leila's mistake?
Choose 1 answer:
(A) Leila evaluated $\cos ^{-1}(0.74)$ in radians instead of in degrees.

B Leila forgot to apply an identity which would reveal another solution in the interval.
(c) Leila forgot to use larger $n$-values which would reveal another solution in the interval.

D Leila gave solutions that were outside of the interval.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the correct interval for the solution. The interval given is from $-360^{\circ}$ to $0^{\circ}$. ‖ ‖ step 2 ⋮ Recognize that $\cos^{-1}(0.74)$ gives the principal value, which is the smallest positive angle whose cosine is 0.74. ‖ ‖ step 3 ⋮ Calculate the principal value: $\cos^{-1}(0.74) = 42.3^{\circ}$. This is the value in the first quadrant. ‖ ‖ step 4 ⋮ Find the second solution in the fourth quadrant using the symmetry of the cosine function: $x = 360^{\circ} - 42.3^{\circ}$. ‖ ‖ step 5 ⋮ Calculate the second solution: $x = 360^{\circ} - 42.3^{\circ} = 317.7^{\circ}$. ‖ ‖ step 6 ⋮ Adjust the second solution to the given interval by subtracting $360^{\circ}$: $x = 317.7^{\circ} - 360^{\circ} = -42.3^{\circ}$. ‖ ‖ step 7 ⋮ Identify that Leila's mistake was not recognizing the second solution in the interval. ‖ ∻1 Answer∻ ***B*** ∻Key Concept∻ ⚹Symmetry of Cosine Function⚹ ∻Explanation∻ ⚹Leila forgot to apply the symmetry of the cosine function to find the second solution in the specified interval. The cosine function is symmetric about the y-axis, meaning that for every positive angle with a certain cosine value, there is a corresponding negative angle within one full rotation (360°) that has the same cosine value.⚹◊What is the correct process for finding the inverse cosine of a given value within a specified interval?⍭ Generate me a similar question◊

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