Conics and Trigonometry

Exploring the Sine

The biggest topic of this course is trigonometry, which involves the functions sine, cosine, and tangent. You may have seen these before in the context of relationships in a right triangle (as shown below), but this course develops them in a fuller sense that allows us to link them to some of the most fundamental things in the universe, like springs, pendulums, light, and sound!

Exploring the Sine

The triangles above are all similar (same shape, different sizes).

Which of these ratios never changes, no matter the size of triangle?

               

Exploring the Sine

We can define the sine of an angle in a right triangle as the length of the side opposite to the angle (marked orange) divided by the hypotenuse (marked purple). We can reduce in symbols using the diagram above to say \( \sin(x) = \frac{O}{H} .\)

If all the marked angles in the right triangles below are congruent, what is the value of \(Q?\)

               

Exploring the Sine

Imagine now that you keep track of the marked blue point as the angle \(x\) increases such that the hypotenuse of the triangle remains a constant length. Part of what kind of shape is traced by the blue point?

               

Exploring the Sine

As the blue point is tracing a circle, we can keep it going and trace a complete circle, from \(0^\circ\) (on the right) to \(90^\circ\) (on the top) all the around back to 360 degrees (on the right again).

The advantage of this is we aren't restricted to a triangle anymore, and we can talk about things like the sine of 180 degrees.

What is the sine of 180 degrees? (Reminder: from the triangle below, \( \sin(x) = \frac{O}{H} .\)

               

Exploring the Sine

If we let the hypotenuse of the triangle equal 1, the sine which is \( \frac{O}{H} \) just becomes \( \frac{O}{1} = O .\) In other words, the length of side opposite to the angle corresponds to the sine.

We can combine this with the previous information from this quiz and think of taking the sine as equivalent of keeping track of the \(y\)-coordinate of a point that rotates around a unit circle (a circle of radius 1) centered on the origin, as shown above.

What does the graph of \( f(x)=\sin(x) \) look like?

               

Exploring the Sine

By turning the graph of the sine into a wave, we have a representation that matches how radio waves, sound waves, light waves, and many more type of waves act.

Pitch is a way to position notes in music that are "lower" or "higher"; the graph above shows a sound wave that starts low in pitch and gradually increases in pitch.

The graph below shows two sound waves. What is true?

(Note: You can hear these waves in the answer!)

               

Exploring the Sine

Waves can be combined to form increasingly complex sounds; even the most complex, multi-instrument music amounts to just a single sound wave that comes from adding multiple waves. In other words, no matter how complex, a sum of sine waves will be able to sound like literally anything.

Later in this course, you’ll learn a lot more about sine (and cosine) waves and about how to work with them algebraically as well as intuitively. If you like, you can play with a pair of sine waves below and see what they look like when combined (the meaning of the different variables will be explored during the course).

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