R I A N G L E

hypotenuse

leg

leg

In a right triangle, the shorter sides are called legs and the longest side (which is the one opposite the right angle) is called the hypotenuse

a

b

c

We’ll label them a, b, and c and the angles and . Trigonometric functions are defined by taking the ratios of sides of a right triangle.

First let’s look at the three basic functions.

SINECOSINE

TANGENTThey are abbreviated using their first 3 letters

ca

hypotenuse

oppositesin

oppositecb

hypotenuse

adjacentcos

adjacent

ba

adjacentoppositetan

We could ask for the trig functions of the angle by using the definitions.

a

b

c

You MUST get them memorized. Here is a mnemonic to help you.

The sacred Jedi word:

SOHCAHTOA

cb

hypotenuse

oppositesin

adjacentcoshypotenuse

ac

oppositetanadjacent

ba

opposite

adjacent

SOHCAHTOA

It is important to note WHICH angle you are talking about when you find the value of the trig function.

a

bc

Let's try finding some trig functions with some numbers. Remember that sides of a right triangle follow the Pythagorean Theorem so

222 cba

Let's choose: 222 5 43 3

45

sin = Use a mnemonic and figure out which sides of the triangle you need for sine.

ho

53

opposite

hypotenuse

tan =

ao

34

opposite

adjacent

Use a mnemonic and figure out which sides of the triangle you need for tangent.

You need to pay attention to which angle you want the trig function of so you know which side is opposite that angle and which side is adjacent to it. The hypotenuse will always be the longest side and will always be opposite the right angle.

This method only applies if you have a right triangle and is only for the acute angles (angles less than 90°) in the triangle.

3

45

Oh, I'm

acute!

So am I!

oppositehypotenusecosecant

hypotenuseoppositesin

There are three more trig functions. They are called the reciprocal functions because they are reciprocals of the first three functions. Oh yeah, this

means to flip the fraction over.

hypotenuseadjacentcos

adjacentoppositetan

adjacenthypotenusesecant

oppositeadjacentcotangent

Like the first three trig functions, these are referred to by the first three letters except for cosecant since it's first three letters are the same as for cosine.

Best way to remember these is learn which is reciprocal of which and flip them.

a

bc

hypotenuse

adjacent is cos

As a way to help keep them straight I think, The "s" doesn't go with "s" and the "c" doesn't go with "c" so if we want secant, it won't be the one that starts with an "s" so it must be the reciprocal of cosine. (have to just remember that tangent & cotangent go together but this will help you with sine and cosine).

3

45

Let's try one: sec =

so

43cot =

45 Which trig function is this the reciprocal of?

adjacenthypotenuse is sec

ha

adjacentopposite istan so

oppositeadjacent iscot

a

o

TRIGONMETRIC IDENTITIESTrig identities are equations that are true for all angles in the domain. We'll be learning lots of them and use them to help us solve trig equations.

RECIPROCAL IDENTITIESThese are based on what we just learned.

sin1cosec

cos1sec

tan1cot

We can discover the quotient identities if we take quotients of sin and cos:

hah0

cossin

Remember to simplify complex fractions you invert and multiply (take the bottom fraction and "flip" it over and multiply to the top fraction).

ah

ho

ao

Which trig function is this?

tan

Try this same thing with and what do you get?

sincos

Now to discover my favorite trig identity, let's start with a right triangle and the Pythagorean Theorem.

Rewrite trading terms places

QUOTIENT IDENTITIESThese are based on what we just learned.

cossintan

sincoscot

a

b c

222 cba 222 cab Divide all terms by c2

c2 c2 c2

1 22

ca

cb Move the exponents to the outside

Look at the triangle and the angle and determine which trig function these are.

oh

This one is sin

a

h

This one is cos

1 cossin 22

1 cossin 22 This is a short-hand way you can write trig functions that are squared

Now to find the two more identities from this famous and often used one.

1 cossin 22 Divide all terms by cos2

cos2 cos2 cos2What trig function is this squared? 1 What trig function

is this squared?

22 sec 1tan

1 cossin 22 Divide all terms by sin2

sin2 sin2 sin2

What trig function is this squared?

1 What trig function is this squared?

22 cosec cot1

These three are sometimes called the Pythagorean Identities since they come from the Pythagorean Theorem

All of the identities we learned are found in the back page of your book under the heading Trigonometric Identities and then Fundamental Identities.

You'll need to have these memorized or be able to derive them for this course.

RECIPROCAL IDENTITIES

sin1cosec

cos1sec

tan1cot

QUOTIENT IDENTITIES

cossintan

sincoscot

22 sec 1tan

22 cosec cot1

PYTHAGOREAN IDENTITIES1 cossin 22

3

If the angle is acute (less than 90°) and you have the value of one of the six trigonometry functions, you can find the other five.

Sine is the ratio of which sides of a right triangle?

Draw a right triangle and label and the sides you know.

When you know 2 sides of a right triangle you can always find the 3rd with the Pythagorean theorem.

a

222 31 a

228 a22

Now find the other trig functions

cosha

223sec

322

Reciprocal of sine so "flip" sine over

cosec 3

tanao

221

"flipped" cos

cot 22 "flipped" tan

31sin

ho

1

There is another method for finding the other 5 trig functions of an acute angle when you know one function. This method is to use fundamental identities.

We'd still get cosec by taking reciprocal of sin

cosec 3Now use my favourite trig identity1cossin 22

Sub in the value of sine that you know

Solve this for cos

98cos2

322

98cos

This matches the answer we got with the other method

You can easily find sec by taking reciprocal of cos.

We won't worry about because angle not negative

square root both sides

31sin

1cos31 2

2

Let's list what we have so far:

cosec 3

We need to get tangent using fundamental identities.

cossintan

Simplify by inverting and multiplying

322cos

Finally you can find cot by taking the reciprocal of this answer.

223sec

223

31

221

22cot

31sin

322

31

tan

SUMMARY OF METHODS FOR FINDING THE REMAINING 5 TRIG FUNCTIONS OF AN ACUTE ANGLE, GIVEN ONE

TRIG FUNCTION.METHOD 1

1. Draw a right triangle labeling and the two sides you know from the given trig function.2. Find the length of the side you don't know by using the Pythagorean Theorem.3. Use the definitions (remembered with a mnemonic) to find other basic trig functions.4. Find reciprocal functions by "flipping" basic trig functions.

METHOD 2Use fundamental trig identities to relate what you know with what you want to find subbing in values you know.

The sum of all of the angles in a triangle always is 180°

ab

c

What is the sum of + ?

Since we have a 90° angle, the sum of the other two angles must also be 90° (since the sum of all three is 180°).

Two angles whose sum is 90° are called

complementary angles.

?sin isWhat ca

?os isWhat cca

adj a

cent

t o

oppo

s it e

adjacent to opposite

Since and are complementary angles and

sin = cos , sine and cosine are called

cofunctions.

This is where we get the name cosine, a cofunction of sine.

90°

Looking at the names of the other trig functions can you guess which ones are cofunctions of each other?

ab

c

Let's see if this is right. Does sec = cosec ?

cosecsec bc

adj a

cent

t o

oppo

s it e

adjacent to opposite

secant and cosecant tangent and cotangent

hypotenuse over adjacent hypotenuse over opposite

This whole idea of the relationship between cofunctions can be stated as:

Cofunctions of complementary angles are equal.

Cofunctions of complementary angles are equal.cos 27°Using the theorem above, what trig function of what angle does this equal?

= sin(90° - 27°) = sin 63°

Let's try one in radians. What trig functions of what angle does this equal?

8tan

82cot

The sum of complementary angles in radians is since 90° is the same as 2

2

83cot

Basically any trig function then equals 90° minus or minus its cofunction.

2

54sin36sin

We can't use fundamental identities if the trig functions are of different angles.

Use the cofunction theorem to change the denominator to its cofunction

36cos36sin

Now that the angles are the same we can use a trig identity to simplify.

36tan

Acknowledgement

I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint.

www.slcc.edu

Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.

Stephen CorcoranHead of MathematicsSt Stephen’s School – Carramarwww.ststephens.wa.edu.au