# help find y.? Relevance
• By the cosine rule,

y^2+37^2-2(37)ycos(42*)=31^2

=>

y^2-54.9927y+408=0

=>

y=46.2 or 8.8

2 possibilities.

• Pinkgreen
Lv 7
6 months agoReport

Explanation: since x is a variable, it may be acute or obtuse, y is therefore has 2 solutions. Using the sine law, sin(x)/37=sin(42*)/31=>
sin(x)=37sin(42*)/31=>sin(x)=0.79863975..=>x=53* or 127*. Thus
y should have 46.2 or 8.8.

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• Law of Sies:

sin A / a = sin B / b = sin C / c

sin 42/31 = sin x/37

Angle ∠C = 85°

The value of y:

46.15245

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• Using the given formula x = 53 degrees and the length of y = 46.2 rounded to the nearest tenth

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• They tell you to use the law of sines

sin(42) / 31 = sin(x) / 37

37 * sin(42) / 31 = sin(x)

x = arcsin(37 * sin(42) / 31)

x + 42 + Y = 180

Y = 180 - 42 - x

Y = 138 - x

Y = 138 - arcsin(37 * sin(42) / 31)

sin(Y) = sin(138 - arcsin(37 * sin(42) / 31))

sin(Y) = sin(138)cos(arcsin(37 * sin(42) / 31)) - sin(arcsin(37 * sin(42) / 31)) * cos(138)

sin(Y) = sin(138) * sqrt(1 - sin(arcsin(37 * sin(42)/31))^2) - cos(138) * (37 * sin(42) / 31)

sin(Y) = sin(138) * sqrt(1 - (37 * sin(42) / 31)^2) - 37 * sin(42) * cos(138) / 31

sin(Y) = sin(138) * (1/31) * sqrt(31^2 - 37^2 * sin(42)^2) - (1/31) * 37 * sin(42) * cos(138)

sin(Y) = (1/31) * (sin(138) * sqrt(961 - 1369 * sin(42)^2) - 37 * cos(138) * sin(42))

sin(Y) / y = sin(42) / 31

y * sin(42) = 31 * sin(Y)

y = 31 * sin(Y) / sin(42)

y = 31 * (1/31) * (sin(138) * sqrt(961 - 1369 * sin(42)^2) - 37 * cos(138) * sin(42)) / sin(42)

sin(138) = sin(42)

y = sqrt(961 - 1369 * sin(42)^2) - 37 * cos(138)

y = 46.152449588397384817605003643097

y = 46.2

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• you have the formula, just use it

sin 42 / 31 = sin x/37

sin x = (37/31) sin 42 = 0.7986

x = 53º

third angle = 180 – 42 – 53 = 85

sin 85/y = sin 42 / 31

y/sin 85 = 31/sin 42

y = 31 sin 85 / sin 42 = 30.882 / 0.6691 = 46.2

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• It's after x and before z.

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• y <--------- there it is

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