Angle ABC is a right triangle and [tex]$\cos \left(22.6^{\circ}\right)=\frac{b}{13}$[/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]?
[tex]$\tan \left(22.6^{\circ}\right)=\frac{a}{13}$[/tex]
[tex]$\tan \left(22.6^0\right)=\frac{13}{a}$[/tex]
[tex]$\tan \left(22.6^{\circ}\right)=\frac{a}{12}$[/tex]
[tex]$\tan \left(22.6^{\circ}\right)=\frac{12}{a}$[/tex]
Answer (2)
Solve for b using the cosine equation: b = 13 ⋅ cos ( 22. 6 ∘ ) ≈ 12 .
Recognize that the tangent function relates the opposite side a and the adjacent side b : tan ( 22. 6 ∘ ) = b a .
Substitute the value of b into the tangent equation: tan ( 22. 6 ∘ ) = 12 a .
Identify the correct equation from the given options: tan ( 22. 6 ∘ ) = 12 a .
Explanation
Problem Analysis We are given a right triangle ABC and the equation cos ( 22. 6 ∘ ) = 13 b . We need to solve for b and round to the nearest whole number. Then, we need to identify the correct equation that uses the value of b to solve for a .
Solve for b First, let's solve for b using the given equation: cos ( 22. 6 ∘ ) = 13 b b = 13 ⋅ cos ( 22. 6 ∘ ) Using a calculator, we find that: b ≈ 13 ⋅ 0.9232 ≈ 12.0017 Rounding to the nearest whole number, we get b = 12 .
Find the correct equation for a Now that we have b = 12 , we need to find the correct equation to solve for a . Since ABC is a right triangle, and we know the angle 22. 6 ∘ and the adjacent side b = 12 , we can use the tangent function to relate the opposite side a to the adjacent side b :
tan ( 22. 6 ∘ ) = b a Substituting b = 12 , we get: tan ( 22. 6 ∘ ) = 12 a
Final Answer Comparing this equation with the given options, we find that the correct equation is: tan ( 22. 6 ∘ ) = 12 a
Examples
Imagine you're building a ramp for skateboarding. You know the angle of the ramp should be 22.6 degrees, and you want the base of the ramp (adjacent side) to be 12 feet long. Using trigonometry, specifically the tangent function, you can calculate the height (opposite side) of the ramp. This ensures the ramp has the correct slope for a smooth ride!
We calculated b using the equation b = 13 ⋅ cos ( 22. 6 ∘ ) ≈ 12 . The correct equation to solve for a is tan ( 22. 6 ∘ ) = 12 a . Therefore, the chosen option is tan ( 22. 6 ∘ ) = 12 a .
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