Question

Please help me solve 1.) solve the right triangle 2.) show the measures of all the sides and angles

Answer 1

Given the ΔABC below:

To solve for a, b and B,

Step 1: Identify the sides of the triangle.

Thus,

`\begin{gathered} AB\Rightarrow hypotenuse\text{ \lparen longest side of the triangle\rparen} \\ BC\Rightarrow opposite\text{ \lparen side facing the angle\rparen} \\ AC\Rightarrow adjacent \end{gathered}`

Step 2: Evaluate a, using trigonometric ratios.

From trigonometric ratios,

`\begin{gathered} \sin\theta=(opposite)/(hypotenuse) \\ \cos\theta=(adjacent)/(hypotenuse) \\ \tan\theta=(opposite)/(adjacent) \end{gathered}`

Thus,

`\begin{gathered} sin\text{ }\theta=(opposite)/(hyotenuse) \\ where \\ \theta\Rightarrow\angle A=36\degree \\ opposite\text{ }\Rightarrow\text{ BC = a} \\ hypotenuse\Rightarrow AB=c=10 \\ thus, \\ \sin36=(a)/(10) \\ cross-multiply, \\ a\text{ = 10}* sin\text{ 36} \\ =10*0.5877852523 \\ \Rightarrow a=5.877852523 \end{gathered}`

Step 3: In a similar manner, evaluate the value of b.

Thus,

`\begin{gathered} \cos\theta=(adjacent)/(hypotenuse) \\ where \\ \theta=36\degree \\ adjacent\Rightarrow AC=b \\ hypotenuse\Rightarrow AB=c=10 \\ thus, \\ \cos36=(b)/(10) \\ cross-multiply, \\ b=10*\cos36 \\ =10*0.8090169944 \\ \Rightarrow b=8.090169944 \end{gathered}`

Step 4: Evaluate the value of B.

`\angle A+\angle B+\angle C=180\degree(sum\text{ of angles in a triangle\rparen}`
`\begin{gathered} where \\ \angle A=36\degree \\ \angle C=90\degree \\ \angle B\text{ is unknown} \\ thus, \\ 36+\angle B+90\text{ =180} \\ \Rightarrow\angle B+126=180 \\ subtract\text{ 126 from both sides of the equation,} \\ \angle B+126-126=180-126 \\ \Rightarrow\angle B=54\degree \end{gathered}`

Hence,

`\begin{gathered} a=5.877852523 \\ b=8.090169944 \\ \angle B=54\degree \end{gathered}`