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Question

Consider a right triangle with a side of length x opposite angle A, a side of length y opposite angle B, and a hypotenuse of length z opposite the right angle. If sinB= 1/√3 and x=8, find the length of the other side (y) and the length of the hypotenuse (z).

 

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Answer 1

sinB = 1/sqr3 = sqr3/3

cosB = sqr6/3 = sinA

B = about 35.26 degrees

A = 54.74 degrees

sinA =sqr6/3 = about 0.8165

sinA/x = sinB/y

sinA/8 = (1/sqr3/y)

sqr6/3)/8 = (sqr3/3)/y

y = 8sqr3/3)/sqr6/3 = (8)sqr(3/6) = (8/sqr2) = 4sqr2 = 5.66

y = 5.66 or exactly 4sqr2

z/sin90 = y/sinB = 5.66/(1/sqr3) = 5.66sqr3 = 5.66(1.732) = 9.798

z = 9.80

z^2 = x^2 + y^2

z^2 = 8^2 + (4sqr2)^2 = 64 +32 = 96

z =sqr96 = 4sqr6 = about 9.8

9.8^2 = 8^2 5.66^2

96.04 = 64 + 32.04

x=8, opposite Angle A

y = 4sqr2 = about 5.66, opposite Angle B

z = 4sqr6 = about 9.80 opposite Angle C = 90 degrees

B= 35.26, A = 54.74, C = 90 degrees

longest side is opposite the largest angle

smallest side is opposite the smallest angle

sinA = sqr3/3

A = sin^-1(sqr3/3) degrees = about 54.74 degrees

B = sin^-2(sqr6/3) degrees = about 35.26 degrees

C = 90 degrees