Triangle [tex]$ABC$[/tex] is a right triangle and [tex]$\cos(22.6^{\circ}) = \frac{b}{13}$[/tex].

Solve for [tex]$b$[/tex] and round to the nearest whole number.

Which equation correctly uses the value of [tex]$b$[/tex] to solve for [tex]$a$[/tex]?

A. [tex]$\tan(22.6^{\circ}) = \frac{a}{13}$[/tex]
B. [tex]$\tan(22.6^{\circ}) = \frac{13}{a}$[/tex]
C. [tex]$\tan(22.6^{\circ}) = \frac{a}{12}$[/tex]
D. [tex]$\tan(22.6^{\circ}) = \frac{12}{a}$[/tex]



Answer :

To solve for [tex]\( b \)[/tex] and find the correct equation for [tex]\( a \)[/tex], follow these detailed steps:

1. Solve for [tex]\( b \)[/tex]:

Given:
[tex]\[ \cos(22.6^\circ) = \frac{b}{13} \][/tex]

We need to isolate [tex]\( b \)[/tex]. Multiply both sides by 13:
[tex]\[ b = 13 \cdot \cos(22.6^\circ) \][/tex]

We know from the given problem that this calculation yields:
[tex]\[ b \approx 12 \][/tex]
when rounded to the nearest whole number.

2. Identify the correct equation for [tex]\( a \)[/tex]:

To determine which equation correctly uses the value of [tex]\( b \)[/tex] to solve for [tex]\( a \)[/tex], we need to use the tangent function.

Since:
[tex]\[ \tan(22.6^\circ) = \frac{\text{opposite}}{\text{adjacent}} \][/tex]

In terms of the triangle [tex]\( \triangle ABC \)[/tex], where angle [tex]\( A \)[/tex] is [tex]\( 22.6^\circ \)[/tex], the sides are:
- Hypotenuse ([tex]\( c \)[/tex]) = 13
- Adjacent side to angle [tex]\( 22.6^\circ \)[/tex] ([tex]\( b \)[/tex]) = 12
- Opposite side to angle [tex]\( 22.6^\circ \)[/tex] ([tex]\( a \)[/tex]) = [tex]\( ? \)[/tex]

The tangent function states:
[tex]\[ \tan(22.6^\circ) = \frac{a}{b} \][/tex]

Substitute [tex]\( b = 12 \)[/tex]:
[tex]\[ \tan(22.6^\circ) = \frac{a}{12} \][/tex]

Therefore, the correct equation that uses the value of [tex]\( b \)[/tex] to solve for [tex]\( a \)[/tex] is:
[tex]\[ \tan \left(22.6^\circ\right) = \frac{a}{12} \][/tex]

So the correct solution steps and the equation giving the value of [tex]\( a \)[/tex] are confirmed. The correct option is:

[tex]\[ \tan \left(22.6^\circ\right) = \frac{a}{12} \][/tex]