How long is its hypotenuse?

Question

The shortest height of a right triangle is 7 cm, and its smallest angle is 15 degrees. How long is its hypotenuse?

Answer 1

sinθ = y / r

sin15 = 7 / r

rsin15 = r(7/r)

rsin15 = 7

rsin15 / sin15 = 7 / sin 15

r = 7 / sin15

r = 7 / 0.258819045

r = 27.04592314 cm

The hypotenuse is 27.05 cm rounded to two decimal places.

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Answer 2

The shortest side of a triangle will be opposite the smallest angle. Therefore the angle oppositie the 7 cm side is 15 degrees. By definition the angle opposite the hypothenuse is the right angle (90 degrees).

Use the law of sines: A / sinA = B / sinB

We get in our problem: 7/ sin15 = hypotheuse / sin 90

since sin90 = 1 the probem becomes hypothenuse = 7/sin15 = 27.0459231...

Answer 3

Use soh. So sin 15 = 7/h because the smallest angle
is opposite the shortest leg.
h = 7/sin 15 = 27.045 cm.
Interestingly, we can get an exact value for h.
Using the half angle formulas(omitting the derivation)
one can prove that sin 15 = (√6 -√2)/4.
So the exact value for h is 28/ (√6 -√2).

Answer 4

the angles of the triangle are 90,15,75.
the largest side is the side opposite to the largest angle,that is 90,obviously is the hypoteneuse.
smalles angle is that opposite to the smallest angle, that is the side opposite to 15.
according to sine rule
sin(15)/smallest=sin(90)/largest
sin(15)/7=sin(90)/x
so x=27.05cm is the hypoteneuse.

Answer 5

smallest angle will be opposite smallest side (we know this from sin rule (a/sinA=b/sinB))

use the first part of SOHCAHTOA
(which means sinx = opp/hyp, cosx=adj/hyp, tanx=opp/adj)

sin(x) = opp/hyp
sin(15) = 7/hyp
hyp = 7/sin(15)

then can use composite angle formula and table of values to get exact answer


sin(x-y) = sinx*cosy - cosx*siny
sin(60-45) = sin(60)*cos(45) - cos (60)*sin(45)
sin(15) = sqrt(3)/2*sqrt(2)/2 - 1/2*sqrt(2)/2
sin(15) = sqrt(6)/4 - sqrt(2)/4
sin(15) = (sqrt(6) - sqrt(2))/4

hyp = 7/sin(15)
hyp = 7/((sqrt(6) - sqrt(2))/4)
hyp = 28/(sqrt(6) - sqrt(2)) cm (exact answer)


28/(√6 - √2) is approx 27.04592314cm but it is best to write it as an exact answer

Answer 6

sin(x) = side opposite / hypotenuse
sin(15) = 7 / hypotenuse
0.258819 = 7 / hypotenuse

hypotenuse = 27.045923 cm or 27.05 cm
.

Answer 7

Hypotenuse = 27.04592314

Answer 8

hyp=shortest side/sin(15

hyp=7/sin(15

hyp=27.04592314cm