Question

WILL MEDAL AND FAN !!!! :)

Answer 1

You know that for any [Image], neither sin[Image] nor cos[Image] can be greater than 1. How can you explain this using the unit circle definitions of sine and cosine? How can you explain it using the right triangle definitions of sine and cosine? As a follow-up question, consider why it is important to have both the right triangle definitions of sine and cosine and the unit circle definitions of sine and cosine. Can you give examples of situations that might be modeled with trigonometric functions? That is, can you give examples of phenomena that take on a series of values over and over again?

I think [image] is supposed to be theta.

Answer 2

Using the unit circle definitions: let (x,y) be the point of intersection between the unit circle and the half-line that makes an angle of theta with the positive x-axis. Then cos(theta) = x and sin(theta) = y. As (x,y) is on the unit circle, the square of the distance from (0,0) is d^2 = 1^2 = 1 = x^2+y^2 = cos^2(theta)+sin^2(theta). Because cos^2(theta) and sin^2(theta) are both non-negative, cos^2(theta) and therefore cos(theta) and sin^2(theta) and therefore sin(theta) must both be less than or equal to 1.

Using the right triangle definitions: let A be the side adjacent to theta, let O be the side opposite to theta and let H be the hypotenuse. Then cos(theta) = A/H and sin(theta) = O/H. The angle opposite to A is equal to 180°-(90°+theta) = 90°-theta ≤ 90°. So, both A and O are opposite to angles ≤ 90° = the angle opposite to H, namely theta and 90°-theta, and so A and O are both ≤ H. From this, it follows that cos(theta) and sin(theta) are always ≤ 1.

The triangle definitions are useful when dealing with triangles or other polygons and with angles between 0° and 90°. The circle definitions are more general and can be used to define cos and sin for all real theta.

A phenomenon that can be modelled using sin and cos is the variation of the displacement from the resting position in a wave over time and the rate of change of this displacement over time. Another example is an oscillating spring.

Note: sin(theta) and cos(theta) can be greater than 1 for complex values of theta.

Answer 3

@antares did you really just type all of that?! Thank you!❤️😩

Answer 4

You're welcome.