Introduction to Introduction to TrigonometryTrigonometry

Right Triangle Trigonometry

Introduction

What special theorem do you already know that applies to a right triangle?

Pythagorean Theorem:a2 + b2 = c2

c

b

a

Introduction

Trigonometry is a branch of mathematics that uses right triangles to help you solve problems.

Trig is useful to surveyors, engineers, navigators, and machinists (and others too.)

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.

Finding Trigonometric Ratios

The three basic trigonometric ratios are sine, cosine, and tangent, which are abbreviated as sin, cos, and tan, respectively.

The word trigonometry is derived from the ancient Greek language and means measurement of triangles.

Before we can understand the trigonometric ratios we need to know how to label Right Triangles.

Labeling Right Triangles

The most important skill you need right The most important skill you need right now is the ability to correctly label the now is the ability to correctly label the sides of a right triangle.sides of a right triangle.

The names of the sides are:The names of the sides are: the the hypotenusehypotenuse the the opposite opposite sideside the the adjacentadjacent side side

Labeling Right Triangles

The hypotenuse is easy to locate because it is always found across from the right angle.

Here is the right angle...

Since this side is across from the right angle, this must be the hypotenuse.

Labeling Right Triangles

Before you label the other two sides you must have a reference angle selected.

It can be either of the two acute angles. In the triangle below, let’s pick angle B as

the reference angle.

A

B

C

This will be our reference angle...

Labeling Right Triangles

Remember, angle B is our reference angle.

The hypotenuse is side BC because it is across from the right angle.

A

B (ref. angle)

C

hypotenuse

Labeling Right Triangles

Side AC is across from our reference angle B. So it is labeled: opposite.

A

B (ref. angle)

Copposite

hypotenuse

Labeling Right Triangles

The only side unnamed is side AB. This must be the adjacent side.

A

B (ref. angle)

C

adjacenthypotenuse

opposite

Adjacent means beside or next to

Labeling Right Triangles

Let’s put it all together. Given that angle B is the reference angle,

here is how you must label the triangle:

A

B (ref. angle)

C

hypotenuse

opposite

adjacent

Labeling Right Triangles

Given the same triangle, how would the sides be labeled if angle C were the reference angle?

Will there be any difference?

Labeling Right Triangles

Angle C is now the reference angle. Side BC is still the hypotenuse since it is

across from the right angle.

A

B

C (ref. angle)

hypotenuse

Labeling Right Triangles

However, side AB is now the side opposite since it is across from angle C.

A

B

C (ref. angle)

oppositehypotenuse

Labeling Right Triangles

That leaves side AC to be labeled as the adjacent side.

A

B

C (ref. angle)adjacent

hypotenuseopposite

Labeling Right Triangles

Let’s put it all together. Given that angle C is the reference

angle, here is how you must label the triangle:

A

B

C (ref. angle)

hypotenuseopposite

adjacent

Labeling Practice

Given that angle X is the reference angle, label all three sides of triangle WXY.

Do this on your own. Click to see the answers when you are ready.

W X

Y

Labeling Practice

How did you do? Click to try another one...

W X

Y

hypotenuse

opposite

adjacent

Labeling Practice

Given that angle R is the reference angle, label the triangle’s sides.

Click to see the correct answers.R

ST

Labeling PracticeLabeling Practice

The answers are shown below:

R

ST

hypotenuse

opposite

adjacent

TRIGONOMETRIC RATIOS

B

CA

h

a

o

hypotenuse sideoppositeA

side adjacent to A

sin A = = oh

side opposite Ahypotenuse

cos A = = ah

side adjacent Ahypotenuse

tan A = =oa

side opposite Aside adjacent to A

The value of the trigonometric ratio depends only on the measure of the acute angle, not on the particular right triangle that is used to compute the value.

A trigonometric ratio is a ratio of the lengths of two sides of a right triangle.

Finding Trigonometric Ratios

Let be a right triangle. The sine, the cosine, and the tangent of the acute angle A are defined as follows.

ABC

Finding Trigonometric Ratios

Compare the sine, the cosine, and the tangent ratios for A in each triangle below.

SOLUTION

The ratios of the sides for a certain angle size stays constant. We can use this to help us find missing sides and missing angles.

Large triangle Small triangle

sin A =opposite

hypotenuse

cos A =adjacent

hypotenuse

tan A =oppositeadjacent

0.47068

17 0.47064

8.5

0.88241517

0.88247.58.5

0.53338

15 0.5333

47.5 Trigonometric ratios

are frequently expressed as decimal approximations.

A

B

C

178

15

A

B

C

8.54

7.5These triangles were created so that A has the same measurement in both triangles.

How do I remember these?

Finding Trigonometric Ratios

Find the sine, the cosine, and the tangent of the indicated angle.

S

R

T S

513

12SOLUTION

The length of the hypotenuse is 13. For S, the length of the opposite side is 5, and the length of the adjacent side is 12.

sin S 0.3846=5

13

cos S 0.9231=1213

tan S 0.4167=5

12

opp.

adj.

hyp.

R

T S

5

12

13

opp.hyp.

=

adj.hyp.

=

opp.adj.

=

Finding Trigonometric Ratios

Find the sine, the cosine, and the tangent of the indicated angle.

R

R

T S

513

12SOLUTION

The length of the hypotenuse is 13. For R, the length of the opposite side is 12, and the length of the adjacent side is 5.

sin R 0.9231=1213

cos R 0.3846=5

13

tan R = 2.4=125

adj.

opp.

hyp.

opp.hyp.

=

adj.hyp.

=

opp.adj.

=

R

T S

5

12

13

Homework

Pg. 469 #3,4,13 Pg. 477 # 3,4,6,7,8