Trigonometry

Definition:

• Trigonometry is the art of studying triangles (in particular, but not limited to, right triangles)

• Trigonometry makes use of both the angles and the side lengths

• Deals with the relationships between the angles and side lengths of a triangle

• The sine, cosine and tangent of acute angles in a 90° triangle show how the side lengths are related to the angles

• Before learning the key formulas in trigonometry, it is of absolute importance that some terms are understood

• Because we are dealing with right triangles, you are already familiar with one very important right triangle theorem:

– The Pythagorean Theorem

a² + b² = c²

• In every right triangle, because one of the angles measures 90°, then logically the other two angles must add up to 90°

A

B C

Because m<B = 90° then m<A + m<C = 90° (since there are 180° in every triangle)

Hypotenuse:– The side that is opposite the

right angle– The longest side in the right

triangle

Opposite Side: – The side that is opposite of a

given angle– Ex: Side AB is opposite m<C

Side BC is opposite m<A

Adjacent Side:– The side that is neither

hypoteneuse or oppositeEx: Side BC is adjacent to m<C

Side AB is adjacent to m<A

A

B C

A

B C

Example:

Fill in the blanks in the following questions:

Hypotenuse: _________________

Opposite m<A: _________________

Adjacent m<A: _________________

Opposite m<C: __________________

Opposite m<C: __________________

These three definitions of the sides are of

utmost importance in trigonometry

They are at the root of finding every angle in a right triangle

Class work and homework:

Hand out 1 and Math 3000 page 182 # 1

Trigonometric Ratios in a Right Triangle

Sine of an acute angle

• The sine of an acute angle is equal to the ratio of the measure of the opposite side to that angle over the measure of the hypotenuse

• The sine of angle A is written sin A

Cosine of an Acute angle

• The cosine of an acute angle is equal to the ratio of the measure of the adjacent side to that angle over the measure of the hypotenuse

• The cosine of angle A is written cos A

Tangent of an acute angle

• The tangent of an acute angle is equal to the ratio of the measure of the opposite side to that angle over the measure of the adjacent side to that angle

• The tangent of angle A is written tan A

The MOST important Gibberish word you will need to remember in

life

SOH – CAH - TOA

A

B C

Adjacent

OppositeA

Hypotenuse

AdjacentA

Hypotenuse

OppositeA

tan

cos

sin

Example:

A

B C

1

2

3

30°

60°

1

360tan

2

160cos

2

360sin

3

130tan

2

330cos

2

130sin

adjacent

opposite

hypotenuse

adjacent

hypotenuse

opposite

• When writing a trigonometric ratio, we can write the measure of the angle when it is known.

• Thus:

• The sine of angle B measuring 30° is written sin 30°

Homework

P. 182 # 1

P. 184 # 2

(2) Review

• Definitions:

• Trig Ratios : SOH CAH TOA

• Hand out 1 – Identifying Opposite, Adjacent, and Hypotenuse & Sine, Cosine, Tangent Problems

• Hand out 2 – Find the Trig Ratio

3. Using your Calculator

Using Your Calculator

1. The keys sin, cos, tan on the calculator enable you to calculate the value of sin A, cos A, or tan A knowing the measure of angle A

So if you know the measure of an angle you can use the sin, cos, or tan buttons on your calculator in order to calculate its value

Formulas for 90° triangle

Formulas to find a missing side Formulas to find a missing angle

(hyp)² = (opp)² + (adj)²

( )

1

Sin A opp

hyp

( )

1

Cos A adj

hyp

( )

1

Tan A opp

adj

1sinopp

Ahyp

1cosadj

Ahyp

1tanopp

Aadj

Summary

• The three new formulas for 90° triangles Sin, Cos, Tan are used to find a missing side length in a right triangle

2. The keys sin-1, cos-1, tan-1 on the calculator enable you to calculate the measure of angle A knowing sin A

• So if know sin A, cos A, or tan A, you can calculate the measure of angle A

Summary

• If we take the inverse of each formula, we can find the missing side angle in a 90° triangle

• The symbol for the inverse of

sin (A) is sin-1; cos (A) is cos-1;

tan (A) is tan-1

Example

sin 30º = 0.5 and sin-1 (0.5) = 30º

Class work

• Mathematics 3000 , page 185 numbers 4 and 5

• Hand out number 2

Finding Missing Angles using Trigonometry Ratios

Formulas for 90° triangle

Formulas to find a missing side Formulas to find a missing angle

(hyp)² = (opp)² + (adj)²

( )

1

Sin A opp

hyp

( )

1

Cos A adj

hyp

( )

1

Tan A opp

adj

1sinopp

Ahyp

1cosadj

Ahyp

1tanopp

Aadj

In a Right Triangle

1. Find the acute angle A when its opposite side and the hypotenuse are known requires the use of sin A

SOH – Opposite/hypotenuse

sin A = M<A=sin-1 ( )=53.1º4

5

4

5

2. Finding the acute angle A when its adjacent side and the hypotenuse are known requires the use of cos A

Cos = adjacent/hypotenuse

Cos A= m<A = cos-1 ( ) = 41.4º 3

43

4

3. Finding the acute angle A when its opposite side and adjacent side are known requires the use of tan A

tan = opposite/adjacent

Tan A = m<A=tan-1 ( ) = 56.3º3

23

2

How to:

1. Label known sides H,O,A

2. Select sine, cosine or tangent depending on information known

3. Set up ratio – leave either as a ratio OR reduce to decimal round to 4 places (thousandth)

4. M<a = inverse of sine, cos or tan and that ratio or decimal

5. Result is your missing angle

Class work

• Mathematics 3000 page 186, activity 4

• Mathematics 3000, page 187, number 8

• Handout number 4

Finding Missing Sides Using

Trigonometric Ratios

Formulas for 90° triangle

Formulas to find a missing side Formulas to find a missing angle

(hyp)² = (opp)² + (adj)²

( )

1

Sin A opp

hyp

( )

1

Cos A adj

hyp

( )

1

Tan A opp

adj

1sinopp

Ahyp

1cosadj

Ahyp

1tanopp

Aadj

In a right triangle

1. Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the hypotenuse, requires the use of sin A

Remember: SOH

*****Cross Multiply*****

Sin 50º= x=5sin50º = 3.83 cm5

x

Finding the measure y of side AC adjacent to the known Angle A, knowing also the measure of the hypotenuse, requires the use of cos A

Remember: cos = adjacent/hypotenuse

*****Cross Multiply*****

Cos 50º = y = 5 cos 50º = 3.21 cm5

y

3. Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the adjacent side to angle A, requires the use of tan A

remember tan=opposite/adjacent

***cross multiply***

tan 30º = x = 4 tan 30º = 2.31 cm4

x

Class work

• Mathematics page 185, activity 3

• Mathematics page 186, numbers 6,7

• Handout number 6

Class Work and Homework

• Page 186, numbers 6 and 7

Solving a triangle

To determine the measure of all its sides and angles

Class work and homework

• Math3000 page187, number 9

• Page 188, number 10, 11,12

Sine Law

• The sides in a triangle are directly proportional to the sine of the opposite angles to these sides

sin sin sin

a b c

A B C

It is also true that:

sin sin sinA B C

a b c

• The sine law can be used to find the measure of a missing side or angle

1st Case

• Finding a side when we know two angles and a side

We calculate the measure x of AC

15 15sin 5013.27

sin 50 sin 60 sin 60

xx cm

How to:

1. Place Measurement x over sin known angle

2. Equal to

3. Measurement known side over sin of known angle

4. Cross multiply and divide to find unknown measurement

5. Calculate.

2nd Case

• Finding an angle when we know two sides and the opposite angle to one of these two sides

• We calculate the measure of angle B

10 13 10sin 50sin 0.5893 36

sin sin 50 13B m B

B

• Make sure you have opposite angles and side measurements. Remember total inside angles must equal 180º

How to calculate if need to find an angle:

1. Place side measurement known over sin of angle we wish to know

2. Equal to

3. side measurement over sin angle we know

4. Cross multiply and divide to find x

5. To calculate angle –sin x = angle. Don’t forget unit i.e.º

Class work and homework

1. Math 3000, page 190, number 1 a and b – we will do altogether

2. Class work: page 190, number 1 c-f and number 2

3. Finish all above work tonight

The sine of an obtuse angle

• The trigonometric functions (sine, cosine, etc.) are defined in a right triangle in terms of an acute angle.  What, then, shall we mean by the sine of an obtuse angle ABC?

• The sine of an obtuse angle is defined to be the sine of its supplement.

• How to find the measure of the degree of an obtuse angle:

• Follow the procedure you have learned so far, then subtract that angle from 180º

18.6 cm

10 cm

22º

10 18.6

sin 22 sin

18.6sin 2210

.697

sin .697

44.2

180 44.2 135.5

inw

• Class work – page 190 #4,6