To solve for [tex]\( b \)[/tex] given that [tex]\( \cos(22.6^\circ) = \frac{b}{13} \)[/tex], and given that the hypotenuse of the right triangle is 13, we can follow these steps:
1. Find [tex]\( b \)[/tex] using the cosine function:
[tex]\[
\cos(22.6^\circ) = \frac{b}{13}
\][/tex]
To isolate [tex]\( b \)[/tex], multiply both sides by 13:
[tex]\[
b = 13 \cdot \cos(22.6^\circ)
\][/tex]
2. The value of [tex]\( \cos(22.6^\circ) \)[/tex] is calculated and then multiplied by 13 to find the precise value of [tex]\( b \)[/tex]:
[tex]\[
b \approx 12.001732822468751
\][/tex]
3. Round [tex]\( b \)[/tex] to the nearest whole number:
[tex]\[
b = 12
\][/tex]
Now that we have found [tex]\( b = 12 \)[/tex], let's determine which equation correctly uses this value to solve for [tex]\( a \)[/tex].
First, recall the definitions of the trigonometric functions:
- Cosine function ([tex]\(\cos\)[/tex]) relates the adjacent side (here [tex]\( b \)[/tex]) to the hypotenuse.
- Tangent function ([tex]\(\tan\)[/tex]) is the ratio of the opposite side to the adjacent side.
Given options:
- Option 1: [tex]\(\tan(22.6^\circ) = \frac{a}{13}\)[/tex]
- Option 2: [tex]\(\tan(22.6^\circ) = \frac{13}{a}\)[/tex]
- Option 3: [tex]\(\tan(22.6^\circ) = \frac{a}{12}\)[/tex]
- Option 4: [tex]\(\tan(22.6^\circ) = \frac{12}{a}\)[/tex]
Let's evaluate each option using the known value of [tex]\( \tan(22.6^\circ) \)[/tex]:
1. Option 1:
[tex]\[
\tan(22.6^\circ) = \frac{a}{13}
\][/tex]
This means we need to check if [tex]\(\tan(22.6^\circ) \approx \frac{a}{13}\)[/tex]. Calculated comparison shows this is False.
2. Option 2:
[tex]\[
\tan(22.6^\circ) = \frac{13}{a}
\][/tex]
This means we need to check if [tex]\(\tan(22.6^\circ) \approx \frac{13}{a}\)[/tex]. Calculated comparison shows this is False.
3. Option 3:
[tex]\[
\tan(22.6^\circ) = \frac{a}{12}
\][/tex]
This means we need to check if [tex]\(\tan(22.6^\circ) \approx \frac{a}{12}\)[/tex]. Calculated comparison shows this is False.
4. Option 4:
[tex]\[
\tan(22.6^\circ) = \frac{12}{a}
\][/tex]
This means we need to check if [tex]\(\tan(22.6^\circ) \approx \frac{12}{a}\)[/tex]. Calculated comparison shows this is False.
After evaluating all the options, we observe that all provided comparisons return False based on the results.
Thus no option correctly correlates [tex]\( \tan(22.6^\circ) \)[/tex] with the given values to solve for [tex]\( a \)[/tex] using the value of [tex]\( b \)[/tex].
