Question

Can somebody please help. FYI Math isn’t my strong suit....please help me!!!!!????

Answer 1

1. Let us first find the missing interior angle.

m<C+53+61=180

m<C=66 degrees

law of sines:
`(sin(A))/(a) =(sin(B))/(b) =(sin(C))/(c)`

a,b, and c represents the lengths of the triangles and A,B, and C represents the measures of the interior angles opposite to the sides.

For the purposes of this problem let us use a shorthand version for the law of sines.

`(sin(B))/(b) =(sin(C))/(c)\\(sin(61degrees))/(b) =(sin(66degrees))/(142)\\b=142sin(61degrees)/sin(66degrees)\\`

b=134.95 meters approx.

AC is equal to length b in this case so AC=134.95 meters approx.

2. law of cosines: a²=b²+c²-2bccos(A)

Let us rearrange this formula so that we can solve for cos(C) in terms of a, b, and c (the sides lengths of the triangle).

a²=b²+c²-2bccos(A)

a²-b²-c²=-2bccos(A)

`(a^(2)-b^(2)-c^(2))/(-2bc)`=cos(A)

cos(A)=
`(a^(2)-b^(2)-c^(2))/(-2bc)`

Now, because we want to find measure of angle A...

cos(A)=
`(13.7^(2)-12.2^(2)-22.1^(2))/(-2(12.2)(22.1))`

cos(A)=0.83 approx.

A=33.52 degrees approx.

3. Solving with law of cosines.

law of cosines: a²=b²+c²-2bccos(A)

a²=16²+18²-2(16)(18)cos(52 degrees)

a²=256+324-576cos(52 degrees)

a²=580-576cos(52 degrees)

a²=225.38 approx.

a=15.01 units approx.

Solving with law of sines.

law of sines:
`(sin(A))/(a) =(sin(B))/(b) =(sin(C))/(c)`

For the purposes of this problem let us use a shorthand version for the law of sines.

`(sin(71 degrees))/(18) =(sin(52 degrees))/(x)\\xsin(71 degrees)=18sin(52degrees)\\x=(18sin(52degrees))/(sin(71 degrees)) \\`

x=15.00 un approx.