Lesson 4 Off on a Tangent Develop Understanding
Jump Start
Find the sine and cosine for the angles of rotation
Learning Focus
Define and identify key features of the tangent graph.
We have extended the definition of sine and cosine to fit angles of rotation. How do we extend the definition of the tangent function?
Open Up the Math: Launch, Explore, Discuss
Recall that the right triangle definition of the tangent ratio is:
1.
Revise this definition to find the tangent of any angle of rotation drawn in standard position, given in either radians or degrees, on any circle of radius
2.
Revise this definition to find the tangent of any angle of rotation drawn in standard position on the unit circle, with
We have observed that on the unit circle the value of sine and cosine can be represented with the length of a line segment.
3.
Indicate on the following diagram which segment’s length represents the value of
a.
Indicate on the diagram which segment’s length represents the value of
b.
There is also a line segment that can be defined on the unit circle so that its length represents the value of
4.
On the coordinate axes below, sketch the graph of
Extend your graph of
Pause and Reflect
5.
Using a unit circle diagram, give exact values for the following trigonometric expressions:
a.
b.
c.
d.
e.
f.
g.
h.
i.
Functions are often classified based on the following definitions:
• A function
• A function
6.
Based on these definitions and your work in this unit, determine how to classify each of the following trigonometric functions.
a.
The function
b.
The function
c.
The function
Pause and Reflect
Ready for More?
When defining the trigonometric ratios using right triangles, we named possible ratios of sides, such as the sine ratio, defined as the ratio of the length of the side opposite the acute angle to the length of the hypotenuse; the cosine ratio, defined as the ratio of the length of the side adjacent to the acute angle to the length of the hypotenuse; and the tangent ratio, defined as the ratio of the length of the side opposite the acute angle to the length of the side adjacent to the acute angle.
It is sometimes useful to consider the reciprocals of these ratios, leading to the definition of three additional reciprocal trigonometric ratios: secant, cosecant, and cotangent, as defined below.
The secant ratio:
The cosecant ratio:
The cotangent ratio:
1.
Complete the following statements:
a.
The ratio is the reciprocal of the sine ratio.
b.
The ratio is the reciprocal of the cosine ratio.
c.
The ratio is the reciprocal of the tangent ratio.
d.
e.
There are also line segments that can be defined on the unit circle so that their lengths represent the value of
Note that
2.
Which segment has a length that would be equal to
3.
Which segment has a length that would be equal to
4.
Which segment has a length that would be equal to
5.
On the coordinate axes, sketch the graph of
6.
On the coordinate axes, sketch the graph of
7.
On the coordinate axes, sketch the graph of
Takeaways
We have extended the definition of sine, cosine, and tangent to include all angles of rotation
On the unit circle, these definitions become:
In the diagram,
This proportion also shows , since
Analyzing the length of
Functions are classified as even or odd, based on the following definitions:
A function is an even function if it satisfies this property:
A function is an odd function if it satisfies this property:
Examining the unit circle for corresponding angles
Vocabulary
- even function
- odd function
- reciprocal trigonometric functions
- vertical asymptote
- Bold terms are new in this lesson.
Lesson Summary
In this lesson, we extended the definition of the tangent ratio for right triangles to include all angles of rotation. Using this definition, we were able to find values for the tangent of angles that are multiples of
1.
The equation and graph of
a.
vertical shift: down
horizontal shift: left
vertical stretch:
Equation:
b.
2.
Explain why it is impossible for