Question

Solve the triangle.

B = 73°, b = 15, c = 10

@phi

Answer 1

can you write down the law of sines for this problem ?

Answer 2

\[\frac{ a }{ \sin A }=\frac{ b }{ \sin B }=\frac{ c }{ \sin C }\]

Answer 3

yes, but we only 2 out of the 3. Leave out the a's
what do we have ?

Answer 4

*need 2 out of 3

Answer 5

\[\frac{ b }{ \sin B }=\frac{ c }{ \sin C }\]?

Answer 6

yes, now replace b, B and c with the numbers they gave you

Answer 7

\[\frac{ 15 }{ \sin 73 }=\frac{ 10 }{ \sin C }\]

Answer 8

now flip the fractions because we want sin C on top
\[ \frac{ \sin 73 }{ 15 }=\frac{ \sin C }{ 10 } \]

multiply both sides by 10. what do you get ?

Answer 9

like this
\[ \frac{ 10 \cdot \sin 73 }{ 15 }= \sin C \]

Answer 10

\[\frac{ 9.563 }{ 150 }= \sin C\]
I think I did it wrong. lol

Answer 11

ok, here is how, in steps
\[ \frac{10}{1} \cdot \frac{ \sin 73 }{ 15 }=\frac{ \cancel{10} \cdot \sin C }{ \cancel{10} }\]

Answer 12

Would it be \[\frac{ 10\sin73 }{ 15 }=\sin C\]or\[\frac{ 9.563 }{ 15 }=\sin C\]

Answer 13

you write down 10* (* for times) on both sides. 10 is 10/1 (to make it a "fraction")
then multiply top * top to get 10 sin73
and bottom times bottom 1*15 = 15

Answer 14

looks good, but divide by 15

Answer 15

It's both of those ways 10 sin73/15 and 9.563/15 or even better 0.6375

Answer 16

we want a number, so we can do the inverse sin to find angle C

Answer 17

Would it be 39.61?

Answer 18

if you mean angle C for "it" yes.

we now have B= 73º, C= 39.61º
all 3 angles in the triangle add up to 180º, so we can figure out angle A. what is it?

Answer 19

A=70.39?

Answer 20

I would add up B and C to get 73+39.61= 112.61
then 180 minus that for the third angle
the check is A+B+C= 180

Answer 21

Oh okay, so A=67.39

67.39+73+39.61=180

Answer 22

Here is how to do it using algebra
A+B+C= 180
replace B and C with their values
A + 73+39.61 = 180
simplify the left side
A + 112.61= 180
add -112.61 to both sides (or subtract 112.61 from both sides)
A + 112.61-112.61= 180-112.61
A= 67.39

Answer 23

what is left is to find side a. Use the Law of Sines again.

Answer 24

\[\frac{ a }{ \sin 67.4 }=\frac{ 15 }{ \sin 73 }\]\[\frac{ \sin 67.4 }{ a }=\frac{ \sin 73 }{ 15 }\]
Do I times it by 15?

Answer 25

I would not flip the fraction, because a is on top. so use the first equation
multiply both sides by sin 67.4 (that means write down sin67.4 don't use a calculator)
don't evaluate (yet), just write the equation.

Answer 26

\[\frac{ a }{ \sin 67.4 }(\frac{ 1 }{ \sin 67.4 })=\frac{ 15 }{ \sin 73 }\left( \frac{ 1 }{ \sin 67.4 } \right)\]\[a=\frac{ 15 }{ \sin73*\sin67.4 }\]
Like this?

Answer 27

almost. but we are not doing magic. you want to multiply by sin 67.4 not 1/sin67.4

notice that
\[ \frac{a}{\sin 67.4} \frac{1}{\sin67.4} \]
the sin * sin does not cancel. you want do divide sin/sin because anything divided by itself is 1

try again.

Answer 28

lost ?

Answer 29

\[\frac{ a }{ \sin67.4 }*\sin67.4\]It'll cancel them and just give us a, right?

Answer 30

\[ \frac{ a }{ \sin 67.4 }(\frac{ \sin 67.4 }{ 1 })=\frac{ 15 }{ \sin 73 }\left( \frac{ \sin 67.4 }{ 1 } \right)\]

Answer 31

yes, that is why we are multiplying both sides by sin 67.4

Answer 32

\[a=\frac{ 15\sin67.4 }{ \sin73 }\]?

Answer 33

yes, now use a calculator to find a

Answer 34

14.5?

Answer 35

yes, that looks good

Answer 36

thank you

Answer 37

Just to be complete, after we found the 2nd angle (angle C)
to be 39.6, we should check for the "ambiguous case"

if there was another solution (there isn't here, but we will check anyway)
the other value of for C would be 180-39.6= 140.4
But when we add angle B = 73 + new angle C 140 we get 213, which is bigger than 180
so this is not an ambiguous case. C cannot be 140 degrees.

But in some problems the 2nd angle 180-C will be small enough...