R I A N G L E

Let's review a few things about inverse functions.

• To have an inverse function, a function must be one-to-one (remember if a horizontal line intersections the graph of a function in more than one place it is NOT one-to-one).

•If we have points on a function graph and we trade x and y places we'll have points on the inverse function graph.

1f f x •Functions and their inverses "undo" each other so

•Since x and y trade places, the domain of the function is the range of the inverse and the range of the function is the domain of the inverse

•The graph of a function and its inverse are reflections about the line y = x (a 45° line).

Is y = sin x a one-to-one function?

1Consider sin

2

Is there more than one that satisfies this equation?

30 and 150 are both solutions

as well as any coterminal angle to these angles.

Since we want a one-to-one function that will have

an inverse we are going to restrict to acute angles.

The following tables list the angles you should know the exact values of. Remember you can get the first table by drawing 30-60-90 or 45-45-90 triangles and the second table contains quadrantal angles

Find the angle that satisfies each equation

where 0 90

2sin

2 2

Since sin 45 , 452

3cos

2

3Since cos30 , 30

2

tan 1 Since tan 45 1, 45

When we are given a function whose input is the trig function of an angle and the output is the angle, it is called an inverse trig function. and is denoted with the inverse function notation. For example:

1 1sin

2

This is called the inverse sine function and means “what angle has a sine value of 1/2?” (we’ll look for an acute angle)30

1cos 1 This is called the inverse cosine function and means “what angle has a cosine value of 1?” (we’ll look for an acute angle)0

1tan 3 60This is called the inverse tangent function and means “what angle has a tangent value of ?” (we’ll look for an acute angle)3

Definition: Inverse Sine, Cosine and Tangent Functions

The reason there is a restriction on the range here is so that this will be a one-to-one function

For help on using your calculator to compute inverse trig functions, click here.

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

TANGENT

They are abbreviated using their first 3 letters

c

a

hypotenuse

oppositesin

oppositec

b

hypotenuse

adjacentcos

adjacent

b

a

adjacent

oppositetan

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 old Indian word:

SOHCAHTOA

c

b

hypotenuse

oppositesin

adjacentcos

hypotenuse

a

c opposite

tanadjacent

b

a

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.

h

o5

3

opposite

hypotenuse

tan =

a

o3

4

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!

opposite

hypotenusecosecant

hypotenuse

oppositesin

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.

hypotenuse

adjacentcos

adjacent

oppositetan

adjacent

hypotenusesecant

opposite

adjacentcotangent

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

4

3cot =

4

5Which trig function is this the reciprocal of?

adjacent

hypotenuse is sec

ha

adjacent

opposite istan so

opposite

adjacent 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.

sin

1csc

cos

1sec

tan

1cot

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

hah0

cos

sin

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

a

h

h

o

a

o

Which trig function is this?

tan

Try this same thing with and what do you get?

sin

cos

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

hah0

cos

sin

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

a

h

h

o

a

o

Which trig function is this?

tan

Try this same thing with and what do you get?

sin

cos

QUOTIENT IDENTITIESThese are based on what we just learned.

cos

sintan

sin

coscot

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.

3

1sin Sine is the ratio of which

sides of a right triangle?h

o

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

1

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

cosh

a

22

3sec3

22

Reciprocal of sine so "flip" sine over

csc 3

tana

o

22

1

"flipped" cos

cot 22"flipped"

tan

RIG

H

RIANGLE

TRIGONOMETRYT

APPLICATIONS

To "solve" a right triangle means to find the measurements all of the angles and lengths of all of the sides. We typically round sides to two decimal places and angles to one.

If we are given at least one of the sides and one of the angles (other than the right angle) or another side, we can solve a right triangle.

We typically label the acute angles and and sides opposite them a and b with the side opposite the right angle (the hypotenuse) c.

a

bc

Solve a right triangle if b = 4 and = 10°

What trig function can we use to relate the angle given, the side given and a side we want to know?

a

bc

410

h

osin

104

Solve for c

10sin

4c 04.23

Use a calculator and make sure it is in degrees.

23.04

The sum of the angles in a triangle is 180° so you know must be 80°.

80

Use the Pythagorean Theorem to find a. 222 04.234 a

22.69

Draw a picture.

c

410sin

Solve a right triangle if b = 4 and = 10°

Could I have used tan instead of sin in the last problem?

a

bc

410

a

otan

104

Solve for a

10tan

4a 69.22

Yes---either way would work.

23.04

The sum of the angles in a triangle is 180° so you know must be 80°.

80

Use the Pythagorean Theorem to find c. 222 69.224 c

22.69

a

410tan

Solve a right triangle if b = 4 and c = 6

What trig function can we use to relate the two sides given, and an angle we want to know?

a

bc

4

h

osin

6

4sin

41.8

4

Find sin-1(4/6)

8.41

6 Use a calculator and make sure it is in degrees.

The sum of the angles in a triangle is 180° so you know must be 180°- 90° - 41.8°.

48.2

Use the Pythagorean Theorem to find a.222 64 a

4.47

6

There are many applications of right triangle trigonometry.

A safety regulation states that the maximum angle of elevation for a rescue ladder is 72°. If a fire department's longest ladder is 110 feet, what is the maximum safe rescue height?

What trig function relates the angle we know, the side we know and the side we want to know?

h

0sin

11072sin

a 72sin110a

ft 62.104

Angle of Elevatio

n

The angle of elevation is measured from a horizontal line looking up at something.

The Angle of Depression is measured from a horizontal line looking down at something.

An outdoor basketball backboard casts a shadow 17 1/3 feet long. The angle of elevation from a point at the end of the shadow to the top of the backboard is 35.8°. Find the height of the backboard.

What would you want to do first? Draw a picture!

17 1/3 ftshadow

h

35.8°

Now you see that this is nothing but a right triangle trig problem. What trig function would you use that relates the angle and side you know to the side you want to know?

31

178.35tan

h

h8.35tan3

117

ft 5.12h