TRIGONOMETRY

http://math.la.asu.edu/~tdalesan/mat170/TRIGONOMETRY.ppt

Angles, Arc length, ConversionsAngle measured in standard position. Angle measured in standard position. Initial side is the positive x – axis which is fixed. Initial side is the positive x – axis which is fixed. Terminal side is the ray in quadrant II, which is free Terminal side is the ray in quadrant II, which is free to rotate about the origin. Counterclockwise rotation to rotate about the origin. Counterclockwise rotation is positive, clockwise rotation is negative.is positive, clockwise rotation is negative.

Coterminal Angles: Angles that have the same terminal side.Coterminal Angles: Angles that have the same terminal side.60°, 420°, and –300° are all coterminal.60°, 420°, and –300° are all coterminal.

Degrees to radians: Multiply angle by Degrees to radians: Multiply angle by .180

318060

radiansradians

Radians to degrees: Multiply angle by Radians to degrees: Multiply angle by .180

45180

4

Arc length = central angle x radius, or Arc length = central angle x radius, or .rs Note: The central angle must be in radian measure.Note: The central angle must be in radian measure.

Note: 1 revolution = 360° = 2π radians. Note: 1 revolution = 360° = 2π radians.

Right Triangle Trig Definitions

• sin(A) = sine of A = opposite / hypotenuse = a/c• cos(A) = cosine of A = adjacent / hypotenuse = b/c• tan(A) = tangent of A = opposite / adjacent = a/b• csc(A) = cosecant of A = hypotenuse / opposite = c/a• sec(A) = secant of A = hypotenuse / adjacent = c/b• cot(A) = cotangent of A = adjacent / opposite = b/a

A

a

b

c

B

C

Special Right Triangles30° 30°

45°

60° 45°

2

11

3 1

1

2

3

3)30tan(

2

1)30sin(

2

3)30cos(

3)60tan(

2

3)60sin(

2

1)60cos(

1)45tan(

2

2)45sin(

2

2)45cos(

Basic Trigonometric Identities

)cos(

)sin()tan(

A

AA

)sin(

)cos()cot(

A

AA

)csc(

1)sin(

)sin(

1)csc(

AA

AA

)sec(

1)cos(

)cos(

1)sec(

AA

AA

)cot(

1)tan(

)tan(

1)cot(

AA

AA

1)(cos)(sin 22 AA)(sec1)(tan 22 AA )(csc)(cot1 22 AA

Quotient identities:

Reciprocal Identities:

Pythagorean Identities:

Even/Odd identities:)csc()csc(

)sin()sin(

AA

AA

)cot()cot(

)tan()tan(

AA

AA

)sec()sec(

)cos()cos(

AA

AA

Even functions Odd functions Odd functions

AAll SStudents TTake CCalculus.Quad II

Quad I

Quad III Quad IV

cos(A)>0sin(A)>0tan(A)>0sec(A)>0csc(A)>0cot(A)>0

cos(A)<0sin(A)>0tan(A)<0sec(A)<0csc(A)>0cot(A)<0

cos(A)<0sin(A)<0tan(A)>0sec(A)<0csc(A)<0cot(A)>0

cos(A)>0sin(A)<0tan(A)<0sec(A)>0csc(A)<0cot(A)<0

Reference AnglesQuad IQuad I

Quad IIQuad II

Quad IIIQuad III Quad IVQuad IV

θ’ = θ θ’ = 180° – θ

θ’ = θ – 180° θ’ = 360° – θ

θ’ = π – θ

θ’ = 2π – θ θ’ = θ – π

Unit circle

• Radius of the circle is 1.

• x = cos(θ)

• y = sin(θ)

• Pythagorean Theorem:

• This gives the identity:

• Zeros of sin(θ) are where n is an integer.

• Zeros of cos(θ) are where n is an integer.

1)sin(1

1)cos(1

122 yx

1)(sin)(cos 22

nn

2

Graphs of sine & cosine

• Fundamental period of sine and cosine is 2π. • Domain of sine and cosine is • Range of sine and cosine is [–|A|+D, |A|+D].• The amplitude of a sine and cosine graph is |A|.• The vertical shift or average value of sine and

cosine graph is D.• The period of sine and cosine graph is • The phase shift or horizontal shift is

DCBxAxg

DCBxAxf

)cos()(

)sin()(

.2

B

.B

C

.

Sine graphs

y = sin(x)

y = sin(3x)

y = 3sin(x)

y = sin(x – 3)

y = sin(x) + 3

y = 3sin(3x-9)+3y = sin(x)

y = sin(x/3)

Graphs of cosine

y = cos(x)

y = cos(3x)

y = cos(x – 3)

y = 3cos(x)y = cos(x) + 3

y = 3cos(3x – 9) + 3y = cos(x)

y = cos(x/3)

Tangent and cotangent graphs

• Fundamental period of tangent and cotangent is π. • Domain of tangent is where n is an integer.• Domain of cotangent where n is an integer.• Range of tangent and cotangent is• The period of tangent or cotangent graph is

DCBxAxg

DCBxAxf

)cot()(

)tan()(

nxx2

|

nxx |

..

B

Graphs of tangent and cotangent

y = tan(x)Vertical asymptotes at

y = cot(x)Verrical asymptotes at .nx .

2nx

Graphs of secant and cosecant

y = sec(x)Vertical asymptotes atRange: (–∞, –1] U [1, ∞) y = cos(x)

y = csc(x)Vertical asymptotes atRange: (–∞, –1] U [1, ∞) y = sin(x)

.2

nx .nx

Inverse Trigonometric Functions and Trig Equations

)arctan()(tan 1 xxy

)arcsin()(sin 1 xxy

)arccos()(cos 1 xxy

2,2

Domain: [–1, 1]

Range:

0 < y < 1, solutions in QI and QII.–1 < y < 0, solutions in QIII and QIV.

Domain: [–1, 1]

Range: [0, π]

0 < y < 1, solutions in QI and QIV.–1< y < 0, solutions in QII and QIII.

2,2

Domain:

Range:

0 < y < 1, solutions in QI and QIII.–1 < y < 0, solutions in QII and QIV.

Trigonometric IdentitiesSummation & Difference Formulas

)tan()tan(1

)tan()tan()tan(

)sin()sin()cos()cos()cos(

)sin()cos()cos()sin()sin(

BA

BABA

BABABA

BABABA

Trigonometric IdentitiesDouble Angle Formulas

)(tan1

)tan(2)2tan(

1)(cos2)(sin21)(sin)(cos)2cos(

)cos()sin(2)2sin(

2

2222

A

AA

AAAAA

AAA

Trigonometric IdentitiesHalf Angle Formulas

)cos(1

)cos(1

2tan

2

)cos(1

2cos

2

)cos(1

2sin

A

AA

AA

AA

The quadrant of 2

A

determines the sign.

Law of Sines & Law of Cosines

)sin()sin()sin(

)sin()sin()sin(

C

c

B

b

A

ac

C

b

B

a

A

)cos(2

)cos(2

)cos(2

222

222

222

Abccba

Baccab

Cabbac

Law of sines Law of cosines

Use when you have a complete ratio: SSA.

Use when you have SAS, SSS.

Vectors• A vector is an object that has a magnitude and a direction.• Given two points P1: and P2: on the plane, a

vector v that connects the points from P1 to P2 is v = i + j.• Unit vectors are vectors of length 1.• i is the unit vector in the x direction.• j is the unit vector in the y direction.• A unit vector in the direction of v is v/||v||• A vector v can be represented in component form by v = vxi + vyj.• The magnitude of v is ||v|| = • Using the angle that the vector makes with x-axis in

standard position and the vector’s magnitude, component form can be written as v = ||v||cos(θ)i + ||v||sin(θ)j

22yx vv

),( 11 yx ),( 22 yx

)( 12 xx )( 12 yy

Vector Operations

Scalar multiplication: A vector can be multiplied by any scalar (or number).Example: Let v = 5i + 4j, k = 7. Then kv = 7(5i + 4j) = 35i + 28j.

Dot Product: Multiplication of two vectors.Let v = vxi + vyj, w = wxi + wyj.

v · w = vxwx + vywy

Example: Let v = 5i + 4j, w = –2i + 3j.v · w = (5)(–2) + (4)(3) = –10 + 12 = 2.

Two vectors v and w are orthogonal (perpendicular) iff v · w = 0.

Addition/subtraction of vectors: Add/subtract same components.Example Let v = 5i + 4j, w = –2i + 3j. v + w = (5i + 4j) + (–2i + 3j) = (5 – 2)i + (4 + 3)j = 3i + 7j.3v – 2w = 3(5i + 4j) – 2(–2i + 3j) = (15i + 12j) + (4i – 6j) = 19i + 6j.||3v – 2w|| = 9.19397619 22

Alternate Dot Product formula v · w = ||v||||w||cos(θ). The angle θ is the angle between the two vectors.

θw

v

Acknowledgements

• Unit Circle: http://www.davidhardison.com/math/trig/unit_circle.gif

• Text: Blitzer, Precalculus Essentials, Pearson Publishing, 2006.