Question

In right triangle ABC, if angle A is 90 degrees, angle C is 34 degrees, and AC is 71.4, find BC.BC = (rounded to the nearest tenth)

Answer 1

From the statement we know that the triangle ABC has:

• angle A = 90°,

,

• angle C = 34°,

,

• side AC = 71.4.

We must compute the length of the side BC.

Given the data above, we draw the following graph of the triangle:

From the graph, we see that the side BC is the hypotenuse of the triangle, and the side AC is the adjacent cathetus with respect to angle C. From trigonometry we have the following identity:

`\cos \theta=(ac)/(h)\text{.}`

Where:

• θ = angle = ,C = 34°,,

,

• ac = adjacent cathethus = ,AC = 71.4,,

,

• h = hyphotenuse = ,BC, (the unknown).

Replacing the data above in the equation, we have:

`\begin{gathered} \cos C=(AC)/(BC), \\ \cos (34^(\circ))=(71.4)/(BC). \end{gathered}`

Solving the last equation for BC, we get:

`\begin{gathered} \cos (34^(\circ))=(71.4)/(BC) \\ BC\cdot\cos (34^(\circ))=71.4, \\ BC=(71.4)/(\cos (34^(\circ)))\cong86.1 \end{gathered}`

Answer

`BC=86.1`