We are asked to use the Law of Sines
to solve the triangle ABC if angle A equals 50 degrees,
side A has a length of 272 units,
and side B has a length of 429 units.
Notice how here, we are given side, side, angle.
When given side, side, angle,
remember, there's a possibility of having no solution,
one solution, or two solutions.
Looking at the triangle below,
I set it up where this angle A,
that measures 50 degrees,
side A is opposite angle A with a length of 272 units.
And I labeled this angle B,
and therefore the opposite side would be side B,
that has a length of 429 units.
And therefore, this would be angle C
with side C opposite angle C.
And notice how I drew this dashed line here,
because we don't know whether this side of 272 units swings
to the right or swings to the left.
And now, let's set up an equation using the Law of Sines
to determine the measure of angle B.
Let's begin with sine A divided by the length of side A,
which gives us sine 50 degrees divided by 272,
must equal the sine of B divided by 429.
And now let's solve the equation for sine B
by multiplying both sides of the equation by 429.
Notice on the right side,
429 divided by 429 simplifies to one,
leaving us with sine B equals.
On the left side,
we have 429 sine 50 degrees
divided by 272.
And now we need to solve this equation for B,
to undo the sine function,
we take the inverse sine of both sides of the equation,
which gives us inverse sine of this quotient,
equals inverse sine
of sine B.
But inverse sine of sine B simplifies perfectly to B,
giving us B is equal to inverse sine
of, again, this quotient.
And now we'll go to the calculator
to determine the measure of angle B.
We first need to make sure we are in degree mode
by pressing the mode key.
And notice how degree is highlighted.
If degree was not highlighted,
we would use the arrows to go down
to degree and press enter.
And now we go back to the home screen
by pressing second mode for quit.
And now we enter inverse sine by pressing second sine,
and now to enter this quotient,
we need the numerator in parentheses.
So we have open parenthesis 429,
sine,
50 degrees,
close parenthesis for the sine function,
close parenthesis for the numerator,
and then divided by 272.
Close parenthesis for the inverse sine function
and enter.
Notice how we get a domain error,
which means it's not possible to create a triangle
from the given information
and we can just say no solution.
And remember,
the value of the sine function is always greater than
or equal to negative one
and less than or equal to positive one,
and this quotient here is either less than one
or greater than one, which is why we get a domain error.
Just to verify this,
let's go ahead and find the value of this quotient.
Let's go ahead and clear this,
and we'll just enter, in parentheses,
429 sine 50 degrees
and then divided by 272, enter.
Notice how the value is greater than one
and there is no angle
that has a sine function value greater than one,
which, again, is the reason why we have no solution.
Notice in the homework,
it does say to enter DNE for all answers
if there is no solution.
So in the homework, be sure to enter DNE here, here,
and here if there is no solution.
I hope you found this helpful.