Question

Find all solutions for a triangle with B=36degrees b=19 c=30

Answer 1

Triangle 1

A =75.86, B=36, C=68.14, b=19, c=30, a=31.34

Triangle 2

A=32.14, B=36, C=111.86, b=19, c=30, a=17.19

Explanation:

We have two sides of triangle (b and c) and an angle (B)

To solve the triangle we use the sine theorem:

`(sin(B))/(b)=(sin(C))/(c) =(sin(A))/(a)`

We substitute the values of b, B and c in the equation and solve for C

`(sin(36))/(19)=(sin(C))/(30)\\\\sin(C) = 30*(sin(36))/(19)\\\\sin(C) = 0.9281\\\\C=arcsin(0.9281)\\\\C=68.14\°\ \ or\ \ C=111.86`

The sum of the internal angles of a triangle is 180, then

`68.14 + 36 + A=180\\\\A=180-68.14 - 36\\\\A=75.86\°`

or

`111.86 + 36 + A=180\\\\A=180-111.86 - 36\\\\A=32.14\°`

No we find "a" for both cases.

`(sin(36))/(19)=(sin(75.86\°))/(a)\\\\0.03094a=sin(75.86\°)\\\\a=(sin(75.86\°))/(0.03094)\\\\a=31.34`

or

`(sin(36))/(19)=(sin(32.14\°))/(a)\\\\0.03094a=sin(32.14\°)\\\\a=(sin(32.14\°))/(0.03094)\\\\a=17.19`

Answer 2

m∠A = 76° , m∠C = 68° , a = 31.4

Explanation:

* Lets revise how to solve a triangle

- In ΔABC

- a, b, c are the lengths of its 3 sides, where

# a is opposite to angle A

# b is opposite to angle B

# c is opposite to angle C

- m∠B = 36°

- b = 19

- c = 30

* To solve the triangle we can use the sin Rule

- In any triangle the ratio between the length of each side

to the measure of each opposite angle are equal

- a/sinA = b/sinB = c/sinC

* Lets use it to find a and m∠C

∵ m∠B = 36° , b = 19 and c = 30

∵ 19/sin36° = 30/sinC ⇒ by using cross multiplication

∴ sinC = 30 × sin36° ÷ 19 = 0.928

∴ m∠C = sin^-1(0.928) = 68°

- Find measure of angle A

∵ The sum of the measures of the interior angles in a triangle is 180°

∵ m∠A = 180° - (68° + 36°) = 180° - 104° = 76°

∴ m∠A = 76°

- Now we can Find a

∵ a/sinA = b/sinB

∴ a/sin76° = 19/sin36° ⇒ by using cross multiplication

∴ a = 19 × sin(76°) ÷ sin(36°) = 31.4

* m∠A = 76° , m∠C = 68° , a = 31.4