Definition II: Right Triangle Trigonometry

TrigonometryMATH 103

S. Rook

Overview

• Section 2.1 in the textbook:– Right triangle Trigonometry– Cofunction theorem– Exact values for common angles

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Right Triangle Trigonometry

Right Triangle Trigonometry

• Another way to view the six trigonometric functions is by referencing a right triangle

• You must memorize the following definition – a helpful mnemonic is SOHCAHTOA:

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hypotenuse

oppositesine

hypotenuse

adjacentcosine

adjacent

oppositetangent

opposite

hypotenuse

sine

1cosecant

adjacent

hypotenuse

cosine

1secant

opposite

adjacent

tangent

1cotangent

Right Triangle Trigonometry (Continued)

• Definition II is an extension of Definition I as long as angle A is acute (why?):– Lay the right triangle on the Cartesian Plane such

that A is the origin and B is the point (x, y)

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r

yA

hyp

A oppsin

r

xA

hyp

A adjcos

x

rA

A adj

hypsec

x

yA

A adj

A opptan

y

rA

A opp

hypcsc

y

xA

A opp

A adjcot

Right Triangle Trigonometry (Example)

Ex 1: For each right triangle ABC, find sin A, csc A, tan A, cos B, sec B, and cot B:

a)

b) C = 90°, a = 3, b = 4

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Cofunction Theorem

Cofunctions

• The six trigonometric functions can be separated into three groups of two based on the prefix co:– sine and cosine– secant and cosecant– tangent and cotangent

• Each of the groups are known as cofunctions• The prefix co means complement or opposite

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Cofunctions and Right Triangles

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Br

yA cos

hyp

B adj

hyp

A oppsin B

r

xA sin

hyp

B opp

hyp

A adjcos

Bx

rA csc

B opp

hyp

A adj

hypsec B

y

rA sec

B adj

hyp

A opp

hypcsc

Bx

yA cot

B opp

B adj

A adj

A opptan B

y

xA tan

B adj

B opp

A opp

A adjcot

Cofunctions and Right Triangles (Continued)

• The measure of the angles in a triangle must sum to 180°

• By definition, a right triangle contains a right angle measuring 90° (C = 90°)

• Therefore, the remaining two angles must sum to 90° (A + B = 90°)

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Cofunction Theorem

• Cofunction Theorem: If angles A and B are complements of each other, then the value of a trigonometric function using angle A will be equivalent to its cofunction using angle B or vice versa

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ABBA

ABBA

ABBA

BA

cottan AND cottan

cscsec AND cscsec

cossin AND cossin

:90 If

Cofunction Theorem (Example)

Ex 2: Use the Cofunction Theorem to fill in the blanks so that each equation becomes a true statement:

a) cot 12° = tan ____b) sec 39° = csc ____c) sin 80° = ___ 10°

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Exact Values for Common Angles

Exact Values for Common Angles

• For select angles, we can obtain exact values for the trigonometric functions:

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2

2

2

1

245sin

x

x

2

2

2

1

245cos

x

x

2

3

2

330cos

x

x2

1

230sin

x

x

2

1

260cos

x

x

2

3

2

360sin

x

x

Exact Values for Common Angles (Continued)

• Only need to memorize the sine and cosine values:– Can derive the remaining trigonometric functions

through identities• e.g.

• Also:

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3

2123

60cos

60sin60tan

10cos 00sin

090cos 190sin undefined0

1

90cos

90sin90tan

Exact Values for Common Angles (Continued)

• In summary, this chart MUST be memorized by chapter 3:

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θ cos θ sin θ

0° 1 0

30°

45°

60°

90° 0 1

2

3

2

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1

2

1

2

2

2

1

2

2

2

1

Exact Values for Common Angles (Example)

Ex 3: For each of the following, replace x with 30°, y with 45°, and z with 60°, and then simplify as much as possible:

a) 3sin(2y)

b) 2sec(90° – z)

c) 4csc(x)17

Summary

• After studying these slides, you should be able to:– Apply the six trigonometric functions to a right triangle– State the definition of a cofunction – Understand and use the Cofunction Theorem– State and use values of the trigonometric functions for

common angles• Additional Practice– See the list of suggested problems for 2.1

• Next lesson– Calculators and Trigonometric Functions of an Acute Angle

(Section 2.2)

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