Two sides and an angle are given below. Determine whether the given information

hrostentsp6

hrostentsp6

Answered question

2021-11-17

Two sides and an angle are given below. Determine whether the given information results in one triangles, or no triangle at all. Solve any resulting triangle(s). 
b=4, c=5, B=30 
Choose the appropriate option below and, if needed, fill in the blanks to complete your selection.
A) A single triangle is produced, where C?, A?, and a? 
B) Two triangles are produced, where the triangle with the smaller angle C has C1?, A1?, and a1?, and the triangle with the larger angle C has C2?, A2?, and a2? 
C) No triangles are produced.

Answer & Explanation

William Yazzie

William Yazzie

Beginner2021-11-18Added 20 answers

Step 1
This problem can be solved using Law of sines of triangle. The Law of Sines is the relationship between the sides and angles of non-right triangles.
Simply, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.

for the triangle ABC shown in figure with side lengths a, b, c the law of sines is given as,
asinA=bsinB=csinC
Step 2
In the question, side b=4, c=5, B=30
applying law of sines,
4sin30=5sinC
40.5=5sinC
8=5sinC
sinC=58=0.625
C=sin1(0.625)=38.68
The sum of angles of triangle =180
therefore
A=18038.6830
A=11.32
now using Law of sines we can find length of third side.
asinA=bsinB
asin111.32=4sin30
a0.931=40.5
a=40.5×0.931=7.45
Step 3
Now let us see if we can have another triangle. We have B=30 given
Now subtract already calculated C from 180
C2=18030141.32
therefore the third angle will be,
A2=18030141.32
A2=8.68
a2sinA=bsinB
a2sin8.68=4sin30
a20.15=40.5
a2=40.5×0.15=1.207
Step 4
Therefore there are two triangles possible for the given measurements b=4, c=5, B=30
Triangle 1
C1=38.68
A1=111.32

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