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Trigonometry

An Overview of

Important Topics

Contents

Trigonometry – An Overview of Important Topics

So I hear you’re going to take a Calculus course? Good idea to brush up on your

Trigonometry!!

Trigonometry is a branch of mathematics that focuses on relationships between

the sides and angles of triangles. The word trigonometry comes from the Latin

derivative of Greek words for triangle (trigonon) and measure (metron).

Trigonometry (Trig) is an intricate piece of other branches of mathematics such

as, Geometry, Algebra, and Calculus.

In this tutorial we will go over the following topics.

 Understand how angles are measured

o Degrees

o Radians

o Unit circle

o Practice

 Solutions

 Use trig functions to find information about right triangles

o Definition of trig ratios and functions

o Find the value of trig functions given an angle measure

o Find a missing side length given an angle measure

o Find an angle measure using trig functions

o Practice

 Solutions

 Use definitions and fundamental Identities of trig functions

o Fundamental Identities

o Sum and Difference Formulas

o Double and Half Angle Formulas

o Product to Sum Formulas

o Sum to Product Formulas

o Law of Sines and Cosines

o Practice

 Solutions

 Understand key features of graphs of trig functions

o Graph of the sine function

o Graph of the cosine function

o Key features of the sine and cosine function

o Graph of the tangent function

o Key features of the tangent function

o Practice

 Solutions

Back to Table of Contents.

Degrees

A circle is comprised of 360°, which is called one revolution

Degrees are used primarily to describe the size of an angle.

The real mathematician is the radian, since most computations are done in

radians.

Radians

1 revolution measured in radians is 2π, where π is the constant approximately

How can we convert between the two you ask?

Easy, since 360° = 2π radians (1 revolution)

Then, 180° = π radians

So that means that 1° =

𝜋

180

radians

And

180

𝜋

degrees = 1 radian

Example 1

Convert 60° into radians

60 ⋅ (1 degree)

𝜋

180

𝜋

180

60 𝜋

180

𝜋

3

radian

Example 2

Convert (-45°) into radians

𝜋

180

− 45 𝜋

180

𝜋

4

radian

Example 3

Convert

3 𝜋

2

radian into degrees

3 𝜋

2

⋅ (1 radian)

180

𝜋

3 𝜋

2

180

𝜋

540 𝜋

2 𝜋

Example 4

Convert −

7 𝜋

3

radian into degrees

−

7 𝜋

3

⋅

180

𝜋

=

1260

3

= 420°

Before we move on to the next section, let’s take a look at the Unit Circle.

Practice Problems

Solutions

Back to Table of Contents.

sec 𝜃 =

𝑟

𝑥

,which is the reciprocal of cos 𝜃

cot 𝜃 =

𝑥

𝑦

, which is the reciprocal of tan 𝜃

You may recall a little something called SOH-CAH-TOA to help your remember the

functions!

SOH… Sine = opposite/hypotenuse

…CAH… Cosine = adjacent/hypotenuse

…TOA Tangent = opposite/adjacent

Example: Find the values of the trigonometric ratios of angle 𝜃

Before we can find the values of the six trig ratios, we need to find the length of

the missing side. Any ideas? Good call, we can use r = √𝑥² + 𝑦² (from the

Pythagorean Theorem)

r =

Now we can find the values of the six trig functions

sin θ =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑡𝑜𝑒𝑛𝑢𝑠𝑒

12

13

csc θ =

ℎ𝑦𝑝𝑡𝑜𝑒𝑛𝑢𝑠𝑒

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

13

12

cos θ =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

5

13

sec θ =

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

13

5

tan θ =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

12

5

cot θ =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

5

12

Example 5

a) Use the triangle below to find the six trig ratios

Example 6

Use the triangle below to find the six trig ratios

Need more help? Click below for a Khan Academy Video

Khan Academy video 2

First use Pythagorean Theorem to find the hypotenuse

a² + b² = c², where a and b are legs of the right triangle and c is the

hypotenuse

6 ² + 8 ² = 𝑐²

36 + 64 = 𝑐²

100 = 𝑐²

√ 100 = √ 𝑐²

10 = 𝑐

sin 𝜃 =

𝑜

ℎ

=

8

10

=

4

5

csc 𝜃 =

1

sin 𝜃

=

5

4

cos 𝜃 =

𝑎

ℎ

=

6

10

=

3

5

sec 𝜃 =

1

cos 𝜃

=

5

3

tan 𝜃 =

𝑜

𝑎

=

8

6

=

4

3

cot 𝜃 =

1

tan 𝜃

=

3

4

1 ² + 𝑏² = ( √ 5 )²

1 + 𝑏² = 5

𝑏² = 4

𝑏 = 2

sin 𝜃 =

2

√ 5

=

2 √ 5

5

csc 𝜃 =

√ 5

2

cos 𝜃 =

1

√ 5

=

√ 5

5

sec 𝜃 =

√ 5

1

= √ 5

tan 𝜃 =

2

1

= 2 cot 𝜃 =

1

2

tan 45° =

sin 45°

cos 45°

√ 2

2

√ 2

2

cot 45° = 1

Example 7

Find sec (

𝜋

4

1

cos(

𝜋

4

)

1

√ 2

2

Example 8

Find tan (

𝜋

6

1

2

√ 3

2

√ 3

3

Example 9

Find cot 240° =

−

1

2

−

√ 3

2

√ 3

3

Using this method limits us to finding trig function values for angles that are

accessible on the unit circle, plus who wants to memorize it!!!

Second Way: If you are given a problem that has an angle measure of 45°, 30°, or

60°, you are in luck! These angle measures belong to special triangles.

If you remember these special triangles you can easily find the ratios for all the

trig functions.

Below are the two special right triangles and their side length ratios

How do we use these special right triangles to find the trig ratios?

If the θ you are given has one of these angle measures it’s easy!

Example 10 Example 11 Example 12

Find sin 30° Find cos 45° Find tan 60°

sin 30° =

1

2

cos 45° =

√ 2

2

tan 60° =

√ 3

1

Third way: This is not only the easiest way, but also this way you can find trig

values for angle measures that are less common. You can use your TI Graphing

calculator.

First make sure your TI Graphing calculator is set to degrees by pressing mode

Find a missing side length given an angle measure

Suppose you are given an angle measure and a side length, can you find the

remaining side lengths?

Yes. You can use the trig functions to formulate an equation to find missing side

lengths of a right triangle.

Example 16

Let’s see another example,

Example 17

Need more help? Click below for a Khan Academy video

Khan Academy video 3

First we know that sin 𝜃 =

𝑜

ℎ

, therefore sin 30 =

𝑥

5

Next we solve for x, 5 ⋅ sin 30 = 𝑥

Use your TI calculator to compute 5 ⋅ sin 30 ,

And you find out 𝑥 = 2. 5

We are given information about the opposite and adjacent

sides of the triangle, so we will use tan

tan 52 =

16

𝑥

𝑥 =

16

tan 52

𝑥 ≈ 12. 5

Find an angle measure using trig functions

Wait a minute, what happens if you have the trig ratio, but you are asked to find

the angles measure? Grab your TI Graphing calculator and notice that above the

sin, cos, and tan buttons, there is 𝑠𝑖𝑛

− 1

− 1

− 1

. These are your inverse

trigonometric functions, also known as arcsine, arccosine, and arctangent. If you

use these buttons in conjunction with your trig ratio, you will get the angle

measure for 𝜃!

Let’s see some examples of this.

Example 18

How about another

Example 19

We know that tan 𝜃 =

8

6

So to find the value of θ, press 2

nd

tan on your calculator

and then type in (8/6)

𝑡𝑎𝑛

− 1

(

8

6

) ≈ 53. 13

𝜃 ≈ 53 .13°

We are given information about the adjacent side and the

hypotenuse, so we will use the cosine function

cos 𝜃 =

1

2

𝑐𝑜𝑠

− 1

(

1

2

) = 60

𝜃 = 60°