Lesson 3 More “Sine” Language Solidify Understanding

Learning Focus

Extend the definition of sine to include all angles of rotation.

How can we define the sine function for angles larger than $90$ °?

Open Up the Math: Launch, Explore, Discuss

Clarita is helping Carlos calculate his height at different locations around a Ferris wheel. They have noticed when they use their formula $h (t) = 30 + 25 \sin (18 t)$ , that their calculator gives them correct answers for the height even when the angle of rotation is greater than $90 °$ . They don’t understand why, since right triangle trigonometry only defines the sine for acute angles.

Carlos and Clarita are making notes of what they have observed about this new way of defining the sine function that seems to be programmed into the calculator.

Carlos: “For some angles, the calculator gives me positive values for the sine of the angle, and for some angles, it gives me negative values.”

1.

Without using your calculator, list at least five angles of rotation for which the value of the sine produced by the calculator should be positive.

2.

Without using your calculator, list at least five angles of rotation for which the value of the sine produced by the calculator should be negative.

Clarita: “Yeah, and sometimes we can’t even draw a triangle at certain positions on the Ferris wheel, but the calculator still gives us values for the sine at those angles of rotation.”

3.

List possible angles of rotation that Clarita is talking about—positions for which you can’t draw a reference triangle. Then, without using your calculator, give the value of the sine that the calculator should provide at those positions.

Carlos: “And, because of the symmetry of the circle, some angles of rotation should have the same values for the sine.”

4.

Without using your calculator, list at least five pairs of angles that should have the same sine value.

Clarita: “Right! And if we go around the circle more than once, the calculator still gives us values for the sine of the angle of rotation, and multiple angles have the same value of the sine.”

5.

Without using your calculator, list at least five sets of multiple angles of rotation where the calculator should produce the same value of the sine.

Carlos: “So how big can the angle of rotation be and still have a sine value?”

Clarita: “Or how small?”

6.

How would you answer Carlos and Clarita’s questions?

Carlos: “And while we are asking questions, I’m wondering how big or how small the value of the sine can be as the angles of rotation get larger and larger?”

7.

Without using a calculator, what would your answer be to Carlos’s question?

Clarita: “Well, whatever the calculator is doing, at least it’s consistent with our right triangle definition of sine as the ratio of the length of the side opposite to the length of the hypotenuse for angles of rotation between $0$ and $90 °$ .”

Carlos and Clarita decide to ask their math teacher how mathematicians have defined sine for angles of rotation, since the ratio definition no longer holds when the angle isn’t part of a right triangle. Here is a summary of that discussion.

We begin with a circle of radius $r$ whose center is located at the origin on a rectangular coordinate grid. We represent an angle of rotation in standard position by placing its vertex at the origin, the initial ray oriented along the positive $x$ -axis, and its terminal ray rotated $θ$ degrees counterclockwise around the origin when $θ$ is positive and clockwise when $θ$ is negative. Let the ordered pair $(x, y)$ represent the point when the terminal ray intersects the circle. (See the diagram, which Clarita diligently copied into her notebook.)

In this diagram, angle $θ$ is between $0$ and $90 °$ ; therefore, the terminal ray is in Quadrant I. A right triangle has been drawn in Quadrant I, similar to the right triangles we have drawn in the Ferris wheel lessons.

8.

Based on this diagram and the right triangle definition of the sine ratio, find an expression for $\sin θ$ in terms of the variables $x$ , $y$ , and $r$ .

$\sin θ =$

9.

Consider the point $(- 3, 4)$ , which is on the circle $x^{2} + y^{2} = 25$ .

a.

What is the radius of this circle?

b.

Draw the circle and the angle of rotation, showing the initial and terminal ray.

c.

For the angle of rotation you just drew, what is the value of the sine based on the definition we wrote for sine in problem 8?

d.

What is the measure of the angle of rotation? How did you determine the size of the angle of rotation?

e.

Is the calculated value based on this definition the same as the value given by the calculator for this angle of rotation?

10.

Consider the point $(- 1, - 3)$ , which is on the circle $x^{2} + y^{2} = 10$ .

a.

What is the radius of this circle?

b.

Draw the circle and the angle of rotation, showing the initial and terminal ray.

c.

For the angle of rotation you just drew, what is the value of the sine based on the definition we wrote for sine in problem 8?

d.

What is the measure of the angle of rotation? How did you determine the size of the angle of rotation?

e.

Is the calculated value based on this definition the same as the value given by the calculator for this angle of rotation?

Ready for More?

In the circle diagram given, draw an angle in standard position that measures between $0$ and $90 degrees$ . Reflect the terminal ray over the $y$ -axis and over the $x$ -axis, and also rotate the terminal ray $180 °$ about the origin.

Find the sine of each of these four angles of rotation.
Find the angle of rotation associated with each of these terminal rays.

Takeaways

In right triangle trigonometry, trigonometric ratios are defined in terms of

For example:

For angles of rotation, trigonometric functions are defined in terms of

For example:

Equations like $h (θ) = 30 + 25 \sin (θ)$ are trigonometric functions since

The statement $\sin (- θ) = - \sin (θ)$ is a true trigonometric identity because

The statement $\sin (θ) = \sin (θ \pm 2 π n)$ is a true trigonometric identity for every natural number $n$ because

The statement $\sin (θ) = \sin (180 - θ)$ is a true trigonometric identity because

Adding Notation, Vocabulary, and Conventions

Annotate the diagram to illustrate the following terms and symbols:

Angle of rotation in standard position
Initial ray
Terminal ray
Point $(x, y)$ on the terminal ray
$θ$
$r$
Coterminal angles

Vocabulary

Lesson Summary

In this lesson, we extended the definition of the sine to make it possible to find sine values for nonacute angles, including all possible angles of rotation $- \infty < θ < \infty$ . We also learned how to draw angles in standard position on a coordinate grid, and to decide if angles are coterminal, and therefore have the same value for the sine function.

Retrieval

1.

Figure 1 shows a funky graph.

a.

Identify each point where there is a maximum and each point where there is a minimum.

b.

This curve repeats itself two and one-half times. (It’s called a periodic function.) Find the length of the interval that would allow you to see exactly one full length of the curve.

c.

The curve is positive on the interval $(- 9, - 3)$ . Identify another interval where this curve is positive.

2.

A $25 ft$ guy line is attached to the top of a carnival tent and anchored in the ground. It makes an angle of $55 °$ with the ground. Find the height of the tent ( $y$ ) and distance the anchor is from the center of the tent ( $x$ ).