Question

Item 23 Solve the triangle with values a=23, b=24, and c=20. Round decimal answers to the nearest tent

Answer 1

Part 1)
`\angle\ C=50.3\°`

Part 2)
`\angle\ B=67.4\°`

Part 3)
`\angle\ A=62.3\°`

Explanation:

step 1

Find the measure of angle C

Applying the law of cosines

`c^(2) =a^(2) +b^(2) -2(a)(b)cos(C)`

substitute the given values and solve for cos(C)

`20^(2) =23^(2) +24^(2) -2(23)(24)cos(C)`

`2(23)(24)cos(C)=23^(2) +24^(2) -20^(2)`

`1,104cos(C)=705`

`cos(C)=705/1,104`

`C=arccos(705/1,104)=50.3\°`

step 2

Find the measure of angle B

Applying the law of cosines

`b^(2) =c^(2) +a^(2) -2(c)(a)cos(B)`

substitute the given values and solve for cos(B)

`24^(2) =20^(2) +23^(2) -2(20)(23)cos(B)`

`2(20)(23)cos(B)=20^(2) +23^(2) -24^(2)`

`920cos(B)=353`

`cos(B)=353/920`

`B=arccos(353/920)=67.4\°`

step 3

Find the measure of angle A

Remember that the sum of the internal angles of triangle must be equal to 180 degrees

`\angle\ A+\angle\ B+\angle\ C=180\°`

substitute the given values and solve for ∠A

`\angle\ A+67.4\°+50.3\°=180\°`

`\angle\ A=180\°-117.7\°`

`\angle\ A=62.3\°`