Lesson 4 Off on a Tangent Develop Understanding

Jump Start

Find the sine and cosine for the angles of rotation $θ$ and $β$ shown in the diagram.

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:

$\tan (A) = \frac{length of side opposite angle A}{length of side adjacent to angle A}$ .

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 $r$ . Explain why your definition is reasonable.

2.

Revise this definition to find the tangent of any angle of rotation drawn in standard position on the unit circle, with $r = 1$ . Explain why your definition is reasonable.

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 $\sin (θ)$ and which represents the value of $\cos (θ)$ for the given angle $θ$ .

a.

Indicate on the diagram which segment’s length represents the value of $\sin (θ)$ and which represents the value of $\cos (θ)$ for the given angle $θ$ .

b.

There is also a line segment that can be defined on the unit circle so that its length represents the value of $\tan (θ)$ . Consider the length of $\overset{―}{D E}$ in the unit circle diagram below. Note that $△ A D E$ and $△ A B C$ are right triangles. Write a convincing argument explaining why the length of segment $D E$ is equivalent to the value of $\tan (θ)$ for the given angle $θ$ .

4.

On the coordinate axes below, sketch the graph of $y = \tan (θ)$ by considering the length of segment $D E$ as $θ$ rotates through angles from $0$ radians to $2 π$ radians. Explain any interesting features you notice in your graph.

Extend your graph of $y = \tan (θ)$ by considering the length of segment $D E$ as $θ$ rotates through negative angles from $0$ radians to $- 2 π$ radians.

Pause and Reflect

5.

Using a unit circle diagram, give exact values for the following trigonometric expressions:

a.

$\tan (\frac{π}{6}) =$

b.

$\tan (\frac{5 π}{6}) =$

c.

$\tan (\frac{7 π}{6}) =$

d.

$\tan (\frac{π}{4}) =$

e.

$\tan (\frac{3 π}{4}) =$

f.

$\tan (\frac{11 π}{6}) =$

g.

$\tan (\frac{π}{2}) =$

h.

$\tan (π) =$

i.

$\tan (\frac{7 π}{3}) =$

Functions are often classified based on the following definitions:

• A function $f (x)$ is classified as an odd function if $f (- x) = - f (x)$ .

• A function $f (x)$ is classified as an even function if $f (- x) = f (x)$ .

6.

Based on these definitions and your work in this unit, determine how to classify each of the following trigonometric functions.

a.

The function $y = \sin (x)$ would be classified as an [an odd function, an even function, neither an odd nor even function]. Give evidence for your response.

b.

The function $y = \cos (x)$ would be classified as an [an odd function, an even function, neither an odd nor even function]. Give evidence for your response.

c.

The function $y = \tan (x)$ would be classified as an [an odd function, an even function, neither an odd nor even function]. Give evidence for your response.

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: $\sec (A) = \frac{length of the hypotenuse}{length of side adjacent to angle A}$

The cosecant ratio: $\csc (A) = \frac{length of the hypotenuse}{length of side opposite angle A}$

The cotangent ratio: $\cot (A) = \frac{length of side adjacent to angle A}{length of side opposite angle A}$

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.

$\frac{1}{\cos θ} =$

$\frac{1}{\sin θ} =$

$\frac{1}{\tan θ} =$

e.

$\frac{1}{\cot θ} =$

$\frac{1}{\sec θ} =$

$\frac{1}{\csc θ} =$

There are also line segments that can be defined on the unit circle so that their lengths represent the value of $\sec (θ)$ , $\csc (θ)$ , or $\cot (θ)$ . These line segments can be used to help us visualize the graphs of the reciprocal trigonometric functions. Consider the lengths of $\overset{―}{A E}$ , $\overset{―}{A G}$ , and $\overset{―}{F G}$ in the unit circle diagram given.

Note that $△ A D E$ and $△ A F G$ are right triangles and $∠ F G A ≅ ∠ D A G$ since they are alternate interior angles formed by parallel lines $\overset{\leftrightarrow}{F G}$ and $\overset{\leftrightarrow}{A D}$ intersected by transversal $\overset{\leftrightarrow}{A G}$ .

2.

Which segment has a length that would be equal to $\sec (θ)$ ? Explain how you know.

3.

Which segment has a length that would be equal to $\csc (θ)$ ? Explain how you know.

4.

Which segment has a length that would be equal to $\cot (θ)$ ? Explain how you know.

5.

On the coordinate axes, sketch the graph of $y = \sec (θ)$ by considering the length of its corresponding segment in the unit circle diagram above as $θ$ rotates through angles from $0$ radians to $2 π$ radians, and from $0$ radians to $- 2 π$ radians. Explain any interesting features you notice in your graph.

6.

On the coordinate axes, sketch the graph of $y = \csc (θ)$ by considering the length of its corresponding segment in the unit circle diagram above as $θ$ rotates through angles from $0$ radians to $2 π$ radians, and from $0$ radians to $- 2 π$ radians. Explain any interesting features you notice in your graph.

7.

On the coordinate axes, sketch the graph of $y = \cot (θ)$ by considering the length of its corresponding segment in the unit circle diagram above as $θ$ rotates through angles from $0$ radians to $2 π$ radians, and from $0$ radians to $- 2 π$ radians. Explain any interesting features you notice in your graph.

Takeaways

We have extended the definition of sine, cosine, and tangent to include all angles of rotation $θ$ using the following definitions:

$\sin θ =$
$\cos θ =$
$\tan θ =$

On the unit circle, these definitions become:

$\sin θ =$
$\cos θ =$
$\tan θ =$

In the diagram, $\tan θ = E D$ , because

This proportion also shows , since

Analyzing the length of $D E$ as angle $θ$ increases around the circle produces the following graph for $f (θ) = \tan θ$ :

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 $θ$ and $- θ$ , the sine function is an odd function because , the cosine function is an even function because , and the tangent function is an odd function because .

Vocabulary

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 $\frac{π}{6}$ or $\frac{π}{4}$ on the unit circle and we were able to determine the key features of the tangent function graph.

Retrieval

1.

The equation and graph of $y = \sqrt[3]{x}$ is given. Write a new equation with the given transformations. Then sketch the new function on the same graph as the parent function.

a.

$y = \sqrt[3]{x}$

vertical shift: down $1$
horizontal shift: left $1$
vertical stretch: $2$

Equation:

b.

2.

Explain why it is impossible for $\sin θ > 1$ .