Understanding sine
A teaching guideline/lesson plan when first teaching sine (grades 7-9)

The sine is simply a RATIO of certain sides of a right triangle.  Look at the triangles below.  They all have the same shape.  That means they have the SAME ANGLES but the lengths of the sides may be different.  In other words, they are SIMILAR figures.

Have your child/students measure the sides s1, h1, s2, h2, s3, h3 as accurately as possible (or draw several similar right triangles on their own).

Those ratios should all be the same (or close to same due to measuring errors).  That is so because the triangles have the same shape (or are similar), which means their respective parts are PROPORTIONAL.  That is why the ratio of those parts remains the same.  Now ask your child what would happen if we had a fourth triangle with the same shape.  The answer of course is that even in that fourth triangle the ratio  s4/h4  would be the same.

The ratio you calculated remains the same for all the triangles. Why? Because the triangles were similar so their sides were proportional. SO, in all right triangles where the angles are the same, this one ratio is the same too. We associate this ratio with the angle α.  THAT RATIO IS CALLED THE SINE OF THE ANGLE α.

What follows is that if you know the ratio, you can find what the angle α is. Or in other words, if you know the sine of α, you can find α. Or, if you know what α is, you can find this ratio - and when you know this ratio and one side of a right triangle, you can find the other sides.

:

s1 h1

=

s2 h2

=

s3 h3

= sin α = 0.57358

In our pictures the angle α is 35 degrees.  So sin 35 = 0.57358 (rounded to five decimals).  We can use this fact when dealing with OTHER right triangles that have a 35 angle.  See, other such triangles are, again, similar to these ones we see here, so the ratio of the opposite side to the hypotenuse, WHICH IS THE SINE OF THE 35 ANGLE,  is the same!  So in another such triangle, if you only know the hypotenuse, you can calculate the opposite side since you know the ratio, or vice versa.

Problem

Suppose we have a triangle that has the same shape as the triangles above.  The side opposite to the 35 angle is 5 cm.  How long is the hypotenuse?

SOLUTION:   Let h be that hypotenuse.  Then

An example

The two triangles are pictured both overlapping and separate.  We can find H3 simply by the fact that these two triangles are similar.  Since the triangles are similar,

We didn't even need the sine to solve that, but note how closely it ties in with similar triangles.

Now we can find the actual angle α from the calculator: 
Since sin α = 0.4333, then α = sin^-10.4333 ≈ 25.7 degrees.

Test your understanding

1.  Draw a right triangle that has a 40 angle.  Then measure the opposite side and the hypotenuse and use those measurements to calculate  sin 40.  Check your answer by plugging into calculator sin 40  (remember the calculator has to be in the degrees mode instead of radians mode).

2.  Draw two right triangles that have a 70 angle - but that are of different sizes.  Use the first triangle to find sin 70 (like you did in problem 1).  Then measure the hypotenuse of your second triangle.  Use sin 70 and the measurement of the hypotenuse to find the opposite side in your second triangle.  Check by measuring the opposite side from your triangle.

3.  Draw a right triangle that has a 48 angle.  Measure the hypotenuse.  Then use sin 48 (from a calculator) and your measurement to calculate the length of the opposite side.  Check by measuring the opposite side from your triangle.

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