How do you connect the concept of sine as the ratio of opposite to hypotenuse with the sine curve?

Question

So I use the unit circle to introduce the sine curve, but I'm always worried a student will ask "But I thought sine was the ratio of the opposite side to the hypotenuse of a right-angled triangle? What *is* sine of 120 degrees? What does that mean?".

And I don't know how to answer that question. I guess it's a gap in my own understanding as much as it is a teaching concern.

How do you introduce the concept of sine?

No student has ever actually asked me that, but I do think I should be able to answer it.

Answer 1

What i do with my students is preface the entire discussion with "our definition from right triangles is not sufficient to use the ratios for non-acute angles. We need to extend to definition in order to get more usefulness from the ratios. However, we need to be careful that our more general definition is still consistent with the more restrictive one". I go on to define sine and cosine as the (x,y) coordinates of the unit circle but make sure to emphasize that the first quadrant is consistent with right triangles. I then make sure my students see that the other three quadrants behave in a way that intuitively feels like proper way to extend the definition(example: we wouldn't like if sin(x)=0 for all x>90 degrees, thats not intuitive, it would be a bad definition. Then we tackle law of sines/cosine for obtuse triangles to give more evidence that the new definition is behaving properly. (Obviously this is a very brief description of what takes multiple class meetings, but its the general idea)

Answer 2

>"But I thought sine was the ratio of the opposite side to the hypotenuse of a right-angled triangle? What is sine of 120 degrees? What does that mean?".

Yes, sine is the ratio of the opposite side to the hypotenuse.  We use a *unit circle* so that the hypotenuse is always one.  This way, the "opposite side" always stays inside the unit circle, so it ranges .

The unit circle is one unit on purpose.  We don't need to write the denominator when it's 1.  (Every number has an invisible denominator of 1.)  For every point along the circle, the sine and cosine values are linked by a2 + b2 =1.  If we know sine, we can find cosine and vice-versa.  Both of them have invisible denominators of 1.  Both of them are bound within the unit circle with values ranging from -1 to +1.

For example, the sine of 30 degrees is 1/2.  That same sine value could be seen on a triangle with a vertical side of 1 and a hypotenuse of 2 (horizontal side is square root of 3).  The same sine value, 1/2 could also be seen inside a unit circle with a vertical side of 1/2 and a hypotenuse of 1 (horizontal side of square root of 3 over 2).  It's the same triangle, just scaled to fit the unit circle so we can all agree on the values as a function of the hypotenuse.

In the calculator, the sine and cosine are the unit circle values, the ratios with a denominator of 1.  Then tangent is the ratio of sine to cosine.  And every other trig function is a form of sine, cosine or combination of both.

Answer 3

That student was me in high school. I remember asking my teacher exactly that question. Despite passing advanced calc 2, I didn't really understand trig until grad school. Maybe this is too simplistic, but I think of sine 120 as being the exact ratio of opposite to hypotenuse that would work for an angle of 120 in a right triangle.

Answer 4

For that aspect, which is where I start, I use similarity.

1) Show that the ratio of side lengths is constant for a given angle in a right triangle, big or small.

2) Talk about how useful a lookup table would be, but what a pain in the butt it would be to do all of that measuring.

3) Explain the historical trigonometrical lookup tables, and how all of that is squeezed into a calculator today.

Answer 5

I typically use special triangles and the hypotenuse to map out the coordinates first. Afterwards we use the trig ratios like regular, made with the triangle of the radius and it's x and y coordinates. To connect the concept of reference angles I typically make the triangle with the x axis. Once there with sine and cosine we can see the hypotenuse, which is the denominator, is always one. This means that the opposite will always be the y coordinates and the adjacent will always be the x coordinates. So on the unit circle sin(theta)=y/1=y and cos(theta)=x/1=x. From here we map all special triangles coordinates(30-60-90 and 45-45-90). From here we can start promting by asking for different values. So when asking what's sin(120°), you can translate this to mean, what is the y value of the terminal side of this angle on the unit circle? Furthermore, you can look at the sine and cosine functions as the y and x values on a circle/ellipse.

So recap: use real points on unit circle as example, use general angle(theta) and x and y coordinates to show the connection between sine, cosine and x ,y of unit circle; make the unit circle with special triangles, then actually use the unit circle to simplify basic trig expressions.

Answer 6

I would tell the student they're right, but they have it backwards. Sine is the y-coordinate(s) from the unit circle. But to show them how it relates to what they know about right triangles, I

1.    Draw a right triangle such that one of the angles is in standard position. I label the sides O, A and H.
2.    If the "tip" of this triangle where on the unit circle, how large would H have to be?
3.    let's assume it's bigger (I draw a little unit circle around the origin).
4.    Let's make a similar triangle that IS on the unit circle. How large does the hypotenuse need to be again? what would I have to multiply/divide the hypotenuse by to make that happen (great chance to try think-pair-share). Eventually, I add a "/H" to the hypotenuse.
5.    We're bending the triangle out of shape, so we need to do the same thing to the other sides to have a similar triangle. I add a "/H" to the O and A sides
6.    What's the length of the hypotenuse now?  If it's one, than "tip" of the triangle is on the unit circle. I draw the first quadrant of the circle.
7.    What are the x and y coordinates of this point (another chance for think-pair-share).
8.    What's the sine of this angle (I point to the angle in standard position).  Eventually we land at the fact that the sine of the angle is length of the side labelled O/H.
9.    I usually wrap up talking about how solving right triangles actually works by similar triangles. "Resizing" the triangle so that it fits on the unit circle.

Answer 7

If you take a right triangle and divide all sides by the length of the hypotenuse, that gets you right back to the unit circle and the opposite side becomes the opposite over hypotenuse

Answer 8

I introduce the unit circle and sin/cos very early, (around 4th or 5th grade) in a "trig boot camp" that takes a few hours. This happens at the time they start programming and I also do a linear algebra boot camp showing basic vector and matrix operations. They see quickly how sin is used to find height of a line and cos the width when drawing lines on the screen.

So there's never any confusion with triangles since the trig functions are introduced in terms of the unit circle and triangles are just a special case of that, rather than sine and cosine needing to be extended later.