Docsity
Docsity

Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity


Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan


Guidelines and tips
Guidelines and tips

Understanding the Unit Circle: Trig Functions, Degrees & Radians, Conversion, Lecture notes of Trigonometry

An overview of the unit circle, its concepts, and its use in calculating trigonometric functions such as sin(θ), cos(θ), tan(θ), sec(θ), csc(θ), and cot(θ). It explains the difference between degrees and radians, and provides conversion factors and examples for converting between the two. The document also introduces the triangle method for solving unit circle problems.

What you will learn

  • How is the unit circle used to calculate trigonometric functions?
  • How can the triangle method be used to solve unit circle problems?
  • What is the difference between degrees and radians?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

sohail
sohail 🇺🇸

4.5

(16)

236 documents

1 / 6

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
11Provided by the Academic Center for Excellence 1 The Unit Circle
Updated October 2019
The Unit Circle
The unit circle can be used to calculate the trigonometric functions sin(θ), cos(θ), tan(θ), sec(θ), csc(θ),
and cot(θ). It utilizes (x,y) coordinates to label the points on the circle, where x represents cos(θ) of a
given angle, y represents sin(θ), and 𝑦
𝑥 represents tan(θ). Theta, or θ, represents the angle in degrees or
radians. This handout will describe unit circle concepts, define degrees and radians, and explain the
conversion process between degrees and radians. It will also demonstrate an additional way of solving
unit circle problems called the triangle method.
What is the unit circle?
The unit circle has a radius of one. The intersection of the x and y-axes (0,0) is known as the origin. The
angles on the unit circle can be in degrees or radians.
The circle is divided into 360 degrees starting on the right side of the xaxis and moving
counterclockwise until a full rotation has been completed. In radians, this would be 2π. The unit circle
is shown on the next page.
Converting Between Degrees and Radians
In trigonometry, most calculations use radians. Therefore, it is important to know how to convert
between degrees and radians using the following conversion factors.
Conversion Factors
𝑫𝒆𝒈𝒓𝒆𝒆𝒔× 𝝅
𝟏𝟖𝟎°=𝑹𝒂𝒅𝒊𝒂𝒏𝒔
𝑹𝒂𝒅𝒊𝒂𝒏𝒔×𝟏𝟖𝟎°
𝝅=𝑫𝒆𝒈𝒓𝒆𝒆𝒔
Degrees
Degrees, denoted by °, are a
measurement of angle size that is
determined by dividing a circle into
360 equal pieces.
Radians
Radians are unit-less but are always
written with respect to π. They
measure an angle in relation to a
section of the unit circle’s
circumference.
Example 1:
Convert 120 to radians.
Step 1: If starting with degrees, 180 should be
on the bottom of the conversion factor so that
the degrees cancel.
120°×𝜋
180°=120°(𝜋)
1(180°)=2𝜋
3
pf3
pf4
pf5

Partial preview of the text

Download Understanding the Unit Circle: Trig Functions, Degrees & Radians, Conversion and more Lecture notes Trigonometry in PDF only on Docsity!

11 Provided by the Academic Center for Excellence 1 The Unit Circle

The Unit Circle

The unit circle can be used to calculate the trigonometric functions sin(θ), cos(θ), tan(θ), sec(θ), csc(θ),

and cot(θ). It utilizes (x,y) coordinates to label the points on the circle, where x represents cos(θ) of a

given angle, y represents sin(θ), and

𝑦

𝑥

represents tan(θ). Theta, or θ, represents the angle in degrees or

radians. This handout will describe unit circle concepts, define degrees and radians, and explain the

conversion process between degrees and radians. It will also demonstrate an additional way of solving

unit circle problems called the triangle method.

What is the unit circle?

The unit circle has a radius of one. The intersection of the x and y-axes (0,0) is known as the origin. The

angles on the unit circle can be in degrees or radians.

The circle is divided into 360 degrees starting on the right side of the x–axis and moving

counterclockwise until a full rotation has been completed. In radians, this would be 2π. The unit circle

is shown on the next page.

Converting Between Degrees and Radians

In trigonometry, most calculations use radians. Therefore, it is important to know how to convert

between degrees and radians using the following conversion factors.

Conversion Factors

𝑫𝒆𝒈𝒓𝒆𝒆𝒔 ×

𝑹𝒂𝒅𝒊𝒂𝒏𝒔 ×

Degrees

Degrees, denoted by °, are a

measurement of angle size that is

determined by dividing a circle into

360 equal pieces.

Radians

Radians are unit-less but are always

written with respect to π. They

measure an angle in relation to a

section of the unit circle’s

circumference.

Example 1:

Convert 120 to radians.

Step 1 : If starting with degrees, 180 should be

on the bottom of the conversion factor so that

the degrees cancel.

120° ×

22 Provided by the Academic Center for Excellence 2 The Unit Circle

The Standard Unit Circle

π

π

2 π

π

π

11π

π

ξ

II I

III

IV

−ξ 2

ξ 2

−ξ 3

ξ

ξ

ξ

ξ

ξ 3

ξ 3

ξ

ξ

ξ

ξ

ξ 3

ξ 3

Y - Axis

X - Axis

Key: (𝐂𝐨𝐬(𝛉), 𝐒𝐢𝐧(𝛉))

44 Provided by the Academic Center for Excellence 4 The Unit Circle

Example 2 :

Use the triangle method to solve:

Step 1: Choose a triangle.

Because 45 is divisible by 45, use the 45°−45°−90° triangle.

Step 2: Draw the triangle in the correct quadrant.

This triangle will be in quadrant I because 45° is between 0 ° and 90°.

Step 3: Analyze the triangle.

Remember that cos(θ) represents

𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝐻𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

. Here, the adjacent side to θ (or 45°) is 1,

and the hypotenuse is ξ

  1. This results in 𝑐𝑜𝑠( 45 °) =

1

ξ

2

Step 4: Rationalize the denominator.

The denominator is rationalized by removing the square roots. Do this by multiplying

the numerator and denominator of the resulting fraction

1

ξ

2

by the radical in the

denominator ξ 2.

Cos

ξ

×

ξ

ξ

Cos

ξ

ξ 2

55 Provided by the Academic Center for Excellence 5 The Unit Circle

Example 3 :

Use the triangle method to solve:

Step 1: Choose a triangle.

Because 240 is divisible by 30, use the 30°−60°−90° triangle.

Step 2: Draw the triangle in the correct quadrant.

This triangle will be in quadrant III because 240 ° is between 18 0° and 270 °. Additionally,

60 ° will be the angle near the origin because 240° is 60° more than 180°.

Step 3: Analyze the triangle.

Note that tan(θ) represents

𝑂𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝐴𝑑𝑗𝑎𝑐𝑒𝑛𝑡

. Here, the opposite side is − ξ

3 while the adjacent

side is − 1. This results in 𝑡𝑎𝑛( 240 °) =

−ξ 3

− 1

Step 4: Simplify.

The negatives cancel each other out to leave

ξ 3

1

, which is ξ 3.

= ξ 3

−ξ 3