How can you 'Solve sin x = 0.5' without using a calculator, and with proof?

Question

I have an upcoming exam, and I am fairly sure this question will crop up in it;

'Solve sinx = 0.5'

Now I know what the answer is. x = 30. But I need to be able to prove it, and I have no idea how!

Answer 1

sin x = 0.5 = 1 / 2
x = 30°

The following may be of use:-

1 is length of side of a right angled triangle
2 is length of hypotenuse of triangle
√3 would then be the remaining side.
Other angles are 60° and 30°
√3 is opposite the 60° angle
From this triangle , the following table may be drawn up:-

Angle___sin___cos___tan
30°____1/2___√3/2__√3/2
60°____√3/2__1/2___√3

Answer 2

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OK. In a right triangle, if one angle is 30 degrees, then the triangle is a 30-60-90 triangle. From geometry, the sides of a 30-60-90 triangle are in the ratio 1 : sqrt(3) : 2. Since the sine is defined as the ratio of opposite side to hypotenuse in a right triangle, it follows that sin(30°)= 1/2 = 0.5. Done! BTW The proof of the ratio of sides in a 30-60-90 triangle is obvious using a geometric proof: Consider an equilateral triangle ABC with side length 2 and with point D as the midpoint of segment BC. Draw an altitude line from A to D. Then ABD is a 30-60-90 triangle with hypotenuse of length 2, and base BD of length 1. The fact that the remaining leg AD has length=sqrt(3) follows directly from the Pythagorean Theorem.

Answer 3

Take an equilateral triangle .The nbisector of one angle divides the opposite side in halves
sin 30 = 1/2side/side =0.5

Answer 4

If you mace a Isopleyro triangle ABC,
with side 2cm . Then if you make
Dichotomos AD you have an
Rectangle triangle ABD whith angle A=30
and sidesAB=2cm, BD=1cm.
then sinA=BD/AB so sinA=1/2
or sin30=0,5

But there are infinity angles x who sinx=0,5 from example sin390=0,5

Answer 5

the solution is x=npi+(-1)^npi/6. [n is any integer]
for proof refer to any standard intermediate level maths text book in the chapter "general solutions of trigonometric equations"

Answer 6

ypu've got to remember your triangles