MAT172 Chapter 7: Trigonometric Functions

Based on Chapter 7: Trigonometric Functions from the 8th edition of Algebra & Trigonometry, Enhanced with Graphing Utilities by Michael Sullivan and Michael Sullivan III. Table of Contents: Section 7.1: cards 1 - 21; Section 7.2: cards 22 - 41; Section 7.3: cards 42 - 43; Section 7.4: cards 44 - 51; Section 7.5: cards 52 - 83; Section 7.6: cards 84 - 102; Section 7.7: cards 103 - 115; Section 7.8: cards 116 - 121.

Degrees to Minutes

1° = 60'

Minutes to Seconds

1’ = 60”

Degree to DMS

Whole degrees, multiply the decimal by 60 and take the whole for minutes, multiply decimals by 60 and take the whole for seconds

DMS to Degree

Divide seconds by 60, add the decimals to minutes and divide by 60, add decimals to degrees

Co-Terminal Angles

Angles that differ by 360°

Positive Angle

Follows counterclockwise rotation

Negative Angle

Follows clockwise rotation

Isosceles Right Triangle

45-90-45

30/60 Triangle

60-90-30

Initial side of an angle

Where the angle begins

Terminal side of an angle

Where the angle stops

Standard position of an angle

When the vertex of an angle is at the origin an the initial side coincides with the positive x-axis

Quadrantal Angle

When the angle is in the standard position and the terminal side lies in one of the quadrants

Arc Length Theorem

s = rθ

Radian measure

θ = s/r

Degree to radians

1° = π/180 radians

Radians to degrees

1 radian = 180/π degrees

Area of a Sector Theorem

A = ½r²θ

Linear Speed of an Object

v = s/t

Angular Speed of an Object

ω = θ/t

Speed of an object traveling in circular motion

v = rω

sinθ

Opposite/hypotenuse

y/r

cosθ

Adjacent/hypotenuse

x/r

tanθ

Opposite/adjacent

y/x

cscθ

Hypotenuse/opposite

r/y

secθ

Hypotenuse/adjacent

r/x

cotθ

Adjacent/opposite

x/y

Reciprocal identity of cscθ

1/sinθ

Reciprocal identity of secθ

1/cosθ

Reciprocal identity of cotθ

1/tanθ

Quotient identity of tanθ

sinθ/cosθ

Quotient identity of cotθ

cosθ/sinθ

sinθ and cosθ Pythagorean identity

sin²θ + cos²θ = 1

cotθ and cscθ Pythagorean identity

1 + cot²θ = csc²θ

tanθ and secθ Pythagorean identity

tan²θ + 1 = sec²θ

sinθ and cosθ Complementary Angle Theorem in degrees

sinθ = cos(90° − θ)

cosθ = sin(90° − θ)

cscθ and secθ Complementary Angle Theorem in degrees

cscθ = sec(90° − θ)

secθ = csc(90° − θ)

tanθ and cotθ Complementary Angle Theorem in degrees

tanθ = cot(90° − θ)

cotθ = tan(90° − θ)

sinθ and cosθ Complementary Angle Theorem in radians

sinθ = cos(π/2 − θ)

cosθ = sin(π/2 − θ)

cscθ and secθ Complementary Angle Theorem in radians

cscθ = sec(π/2 − θ)

secθ = csc(π/2 − θ)

tanθ and cotθ Complementary Angle Theorem in radians

tanθ = cot(π/2 − θ)

cotθ = tan(π/2 − θ)

Angle of elevation

The acute angle measured from the horizontal to line-of-sight observation of the object when looking up

Angle of depression

The acute angle measured from the horizontal to line-of-sight observation of the object when looking down

Co-terminal Angles

Two angles in standard position that have the same terminal side

What quadrants are sinθ and cscθ positive?

Quadrant I and Quadrant II

What quadrants are sinθ and cscθ negative?

Quadrant III and Quadrant IV

What quadrants are cosθ and secθ positive?

Quadrant I and Quadrant IV

What quadrants are cosθ and secθ negative?

Quadrant II and Quadrant III

What quadrants are tanθ and cotθ positive?

Quadrant I and Quadrant III

What quadrants are tanθ and cotθ negative?

Quadrant II and Quadrant IV

Reference Angle

The acute angle formed by the terminal side of θ and the x-axis

What is the point (x, y) in terms of cosθ and sinθ?

(cos t, sin t)

What is the sinθ of the point (x, y) in terms of x, y, r?

sinθ = y/r

What is the cosθ of the point (x, y) in terms of x, y, r?

cosθ = x/r

What is the tanθ of the point (x, y) in terms of x, y, r?

tanθ = y/x, x ≠ 0

What is the cscθ of the point (x, y) in terms of x, y, r?

cscθ = r/y, y ≠ 0

What is the secθ of the point (x, y) in terms of x, y, r?

secθ = r/x, x ≠ 0

What is the cotθ of the point (x, y) in terms of x, y, r?

cotθ = x/y, y ≠ 0

What is the domain of sinθ?

All real numbers

What is the range of sinθ?

-1 ≤ sinθ ≤ 1

What is the domain of cosθ?

All real numbers

What is the range of cosθ?

-1 ≤ cosθ ≤ 1

What is the domain of tanθ?

All real numbers except odd multiples of π/2 (π/2, 3π/2, 5π/2 …)

What is the range of tanθ?

-∞ < tanθ < ∞

What is the domain of cotθ?

All real numbers except integer multiples of π (0, π, 2π, 3π …)

What is the range of cotθ?

-∞ < cotθ < ∞

What is the domain of secθ?

All real numbers except odd multiples of π/2 (π/2, 3π/2, 5π/2 …)

What is the range of secθ?

secθ ≤ -1 or secθ ≥ 1

What is the domain of cscθ?

All real numbers except integer multiples of π (0, π, 2π, 3π …)

What is the range of cscθ?

cscθ ≤ -1 or cscθ ≥ 1

When is a function called periodic?

A function ƒ is called periodic if there is a positive number p with the property that, whenever θ is in the domain of ƒ, so is θ + p, and ƒ(θ + p) = ƒ(θ)

What is the period of sinθ?

2π or 360°

What is the period of cosθ?

2π or 360°

What is the period of tanθ?

π or 180°

What is the period of cscθ?

2π or 360°

What is the period of secθ?

2π or 360°

What is the period of cotθ?

π or 180°

What is the even-odd property of sinθ?

sin(-θ) = -sinθ

What is the even-odd property of cosθ?

cos(-θ) = cosθ

What is the even-odd property of tanθ?

tan(-θ) = -tanθ

What is the even-odd property of cscθ?

csc(-θ) = -cscθ

What is the even-odd property of secθ?

sec(-θ) = secθ

What is the even-odd property of cotθ?

cot(-θ) = -cotθ

Function of sinθ in an xy-plane

ƒ(x) = sin x

Function of cosθ in an xy-plane

ƒ(x) = cos x

Function of tanθ in an xy-plane

ƒ(x) = tan x

Function of cscθ in an xy-plane

ƒ(x) = csc x

Function of secθ in an xy-plane

ƒ(x) = sec x

Function of cotθ in an xy-plane

ƒ(x) = cot x

Draw a graph of ƒ(x) = sin x

What are the intercepts of ƒ(x) = sin x?

x = … -2π, -π, 0, π, 2π, …

y = 0

What is the maximum of ƒ(x) = sin x?

y = 1 at x = … -3π/2, π/2, 5π/2, …

What is the minimum of ƒ(x) = sin x?

y = -1 at x = … -π/2, 3π/2, 7π/2, …

Draw a graph of ƒ(x) = cos x

What are the intercepts of ƒ(x) = cos x?

x = … -3π/2, -π/2, π/2, 3π/2, …

y = 1

What is the maximum of ƒ(x) = cos x?

y = 1 at x = … -2π, 0, π, …

What is the minimum of ƒ(x) = cos x?

y = -1 at x = -π, π, 3π, …

What are sinusoidal functions?

Any curve that is a transformation of the sine curve

Sinusoidal relationship between cos x and sin x

cos(x - π/2) = sin x

What is the amplitude of a sinusoidal function?

The height between the max/min value and x-axis, written as A in ƒ(x) = A sin x