Question

Math

Posted 5 months ago

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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.
```

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Answer from Sia

Posted 5 months ago

Solution by Steps

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.

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