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TRIGONOMETRY

- http//math.la.asu.edu/tdalesan/mat170/TRIGONOMET

RY.ppt

Angles, Arc length, Conversions

Angle measured in standard position. Initial

side is the positive x axis which is fixed.

Terminal side is the ray in quadrant II, which

is free to rotate about the origin.

Counterclockwise rotation is positive, clockwise

rotation is negative.

Coterminal Angles Angles that have the same

terminal side. 60, 420, and 300 are all

coterminal.

Degrees to radians Multiply angle by

radians

Radians to degrees Multiply angle by

Note 1 revolution 360 2p radians.

Arc length central angle x radius, or

Note The central angle must be in radian measure.

Right Triangle Trig Definitions

B

c

a

A

C

b

- 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

Special Right Triangles

30

45

2

1

60

45

1

1

Basic Trigonometric Identities

Quotient identities

Even/Odd identities

Even functions

Odd functions

Odd functions

Reciprocal Identities

Pythagorean Identities

All Students Take Calculus.

Quad I

- Quad II

cos(A)gt0 sin(A)gt0 tan(A)gt0 sec(A)gt0 csc(A)gt0 cot(A

)gt0

cos(A)lt0 sin(A)gt0 tan(A)lt0 sec(A)lt0 csc(A)gt0 cot(A

)lt0

cos(A)lt0 sin(A)lt0 tan(A)gt0 sec(A)lt0 csc(A)lt0 cot(A

)gt0

cos(A)gt0 sin(A)lt0 tan(A)lt0 sec(A)gt0 csc(A)lt0 cot(A

)lt0

Quad IV

Quad III

Reference Angles

Quad I

Quad II

? ?

? 180 ?

? p ?

? ? 180

? 360 ?

? 2p ?

? ? p

Quad III

Quad IV

Unit circle

- Radius of the circle is 1.
- x cos(?)
- y sin(?)
- Pythagorean Theorem
- This gives the identity
- Zeros of sin(?) are where n is an integer.
- Zeros of cos(?) are where n is an integer.

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Graphs of sine cosine

- Fundamental period of sine and cosine is 2p.
- Domain of sine and cosine is
- Range of sine and cosine is AD, AD.
- 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

Sine graphs

y sin(x)

y sin(x) 3

y 3sin(x)

y sin(3x)

y sin(x 3)

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

y sin(x/3)

Graphs of cosine

y cos(x)

y cos(x) 3

y 3cos(x)

y cos(3x)

y cos(x 3)

y 3cos(3x 9) 3 y cos(x)

y cos(x/3)

Tangent and cotangent graphs

- Fundamental period of tangent and cotangent is p.

- 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

Graphs of tangent and cotangent

y tan(x) Vertical asymptotes at

y cot(x) Verrical asymptotes at

Graphs of secant and cosecant

y csc(x) Vertical asymptotes at Range (8, 1

U 1, 8) y sin(x)

y sec(x) Vertical asymptotes at Range (8, 1

U 1, 8) y cos(x)

Inverse Trigonometric Functions and Trig Equations

Domain 1, 1 Range

0 lt y lt 1, solutions in QI and QII. 1 lt y lt 0,

solutions in QIII and QIV.

Domain Range

Domain 1, 1 Range 0, p

0 lt y lt 1, solutions in QI and QIV. 1lt y lt 0,

solutions in QII and QIII.

0 lt y lt 1, solutions in QI and QIII. 1 lt y lt 0,

solutions in QII and QIV.

Trigonometric IdentitiesSummation Difference

Formulas

Trigonometric IdentitiesDouble Angle Formulas

Trigonometric IdentitiesHalf Angle Formulas

The quadrant of

determines the sign.

Law of Sines Law of Cosines

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 vectors magnitude,

component form can be written as v vcos(?)i

vsin(?)j

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.

Alternate Dot Product formula v w

vwcos(?). The angle ? is the angle

between the two vectors.

v

?

w

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

Acknowledgements

- Unit Circle http//www.davidhardison.com/math/tri

g/unit_circle.gif - Text Blitzer, Precalculus Essentials, Pearson

Publishing, 2006.