Section 7.1: Angles and Their Measures

• Def: A ray is a portion of a line that starts at a point V on the line andextends indefinitely in one direction. The vertex of a ray is the startingpoint, V , of the ray.

• Def: If two rays are drawn with a common vertex then the angle between thetwo rays is a measure of how much you have to rotate one ray to get to theother ray. When there is an angle between two rays, the ray you start at iscalled the initial side and the ray you end at, after rotating the initial side,is the terminal side.

• Notation: Angles are usually symbolized by Greek letters, such as α, β, γ, θ.

• Note: The angle between two rays is not uniquely measured. You can rotatethe initial side clockwise or counterclockwise to get to the terminal side. Also,you can rotate the initial side directly to the terminal side or you can rotatethe initial side through one full rotation before rotating it to the terminal sideor you can rotate the initial side through two full rotations before rotating itto the terminal side, etc.

• Def: An angle θ is said to be in standard position if the vertex is at the originand its initial side is along the positive side of the x-axis.

• If the terminal side of an angle θ lies in one of the four quadrants, then wesay that θ lies in the quadrant in which the terminal side lies. If the terminalside of θ lies on the x- or y-axis, then we say that θ is a quadrantal angle.For example, in the first figure below, θ is in quadrant III and in the secondfigure, θ is a quadrantal angle.

1

• One of the two most common units used to measure angles is degrees. If youwere to rotate the initial side of an angle through one full revolution (so theterminal side lies on top of the initial side), the angle would be 360◦. Thus,each degree is 1

360of a revolution; i.e., 1◦ = 1

360revolutions.

• Def: A right angle is an angle measuring 90◦. A straight angle is an anglemeasuring 180◦.

• ex. Draw each angle:

(a) 120◦

(b) 270◦

(c) −45◦

(d) −405◦

• Def: A central angle is a positive angle whose vertex is the center of a circle.The rays of a central angle intersect an arc on the circle.

2

• The other of the two most common units used to measure angles is radians.One radian is defined to be the measure of the central angle whose raysintersect the circle of radius r to form an arc of length r.

• Theorem: The length, s, of the arc on the circle of radius r which is intersectedby a central angle of θ radians is

s = rθ.

• ex. Find the missing angle:

(a) r = 10 m, θ = 3 rad

(b) s = 4 ft, θ = 25

rad

• Relationship between degrees and radians: For a full revolution around a cir-cle of radius r, s = 2πr (if we go one full revolution then the initial and ter-minal sides lie on top of each other, so the arc between them is the circle itselfand the length of the circle is the circumference of the circle). So, 2πr = rθso θ = 2π. So, one revolution is 2π radians. But in terms of degrees, onerevolution is 360◦, so 2π rad = 360◦, so π rad = 180◦. Thus, we have thefollowing relations:

(i) 1◦ = π180

rad

3

(ii) 1 rad = 180◦

π

• ex. Convert each angle in degrees to radians. Express your answer as amultiple of π.

(a) 45◦

(b) 135◦

(c) −210◦

(d) −450◦

• Convert each angle in radians to degrees.

(a) π4

(b) −7π4

(c) −3π

• ex. Draw the angle 9π4

.

• ex. Find the missing quantity: r = 8 in, θ = 45◦

4

Section 7.2: Right Triangle Trigonometry

• Def: The trigonometric functions of acute angles are the six ratios which canbe obtained from a right triangle. The six trigonometric functions are definedas follows:

1. sine of θ = sin θ = bc

= oppositehypotenuse

2. cosine of θ = cos θ = ac

= adjacenthypotenuse

3. tangent of θ = tan θ = ba

= oppositeadjacent

4. cosecant of θ = csc θ = cb

= hypotenuseopposite

5. secant of θ = sec θ = ca

= hypotenuseadjacent

6. cotangent of θ = cot θ = ab

= adjacentopposite

• ex. Find the value of the six trigonometric functions of the angle θ in thefigure.

• Among the six trigonometric functions, there are some relationships betweensome of them.

– Reciprocal Identities:

csc θ = 1sin θ

sec θ = 1cos θ

cot θ = 1tan θ

sin θ = 1csc θ

cos θ = 1sec θ

tan θ = 1cot θ

1

– Quotient Identities:

tan θ = sin θcos θ

cot θ = cos θsin θ

• ex. Use the definition or identities to find the exact value of each of theremaining five trigonometric functions of the acute angle θ.

(a) cos θ =√

24

(b) cot θ = 3

• Pythagorean Identities:

sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ cot2 θ + 1 = csc2 θ

• Note: The second and third of the Pythagorean identities are obtained fromthe first identity by dividing each term by either cos2 θ or by sin2 θ, respec-tively, and using the reciprocal or quotient identities to simplify.

• Collectively, the reciprocal identities, the quotient identities, and the Pythagoreanidentities are called the Fundamental identities.

• Def: Two acute angles of a right triangle are called complementary if theirsum is 90◦. In the diagram below, the angles α and β are complementaryangles.

2

• Note: For the complementary angles α and β, the following relationshipsbetween the trigonometric functions exist:

sin α = bc

= cos β cos α = ac

= sin β tan α = ba

= cot β

csc α = cb

= sec β sec α = ca

= csc β cot α = ab

= tan β

• Def: Trigonometric functions which are related by having the same valueat complementary angles are called cofunctions. Thus, sine and cosine arecofunctions, cosecant and secant are cofunctions, and tangent and cotangentare cofunctions.

• Complementary Angle Theorem: Cofunctions of complementary angles areequal.

• The Complementary Angle Theorem just says in words what the relationshipsbetween the trigonometric functions of complementary angles above say inequations.

• Another way of stating the Complementary Angle Theorem is given by thefollowing relationships (each relation is stated for θ given in degrees or inradians):

sin θ = cos (90◦ − θ) sin θ = cos(π2− θ

)cos θ = sin (90◦ − θ) cos θ = sin

(π2− θ

)tan θ = cot (90◦ − θ) tan θ = cot

(π2− θ

)csc θ = sec (90◦ − θ) csc θ = sec

(π2− θ

)sec θ = csc (90◦ − θ) sec θ = csc

(π2− θ

)cot θ = tan (90◦ − θ) cot θ = tan

(π2− θ

)• Use Fundamental Identities and/or the Complementary Angle Theorem to

find the exact value of each expression.

(a) csc2 40◦ − cot2 40◦

3

(b) sin 38◦

cos 52◦

(c) sin 40◦ · csc 50◦ · cot 40◦

• ex. Given cos 60◦ =√

32

, use trigonometric identities to find the exact valueof

(a) sin 30◦

(b) sin2 60◦

(c) csc π6

(d) sec π3

4

Section 7.3: Computing the Values of TrigonometricFunctions of Acute Angles

• The value of the six trigonometric functions for certain acute angles fromgeometry are easy to figure out. For sin θ and cos θ we have the followingvalues:

θ (Degrees) θ (Radians) sin θ cos θ

30◦ π6

12

√3

2

45◦ π4

√2

2

√2

2

60◦ π3

√3

212

The values of the remaining trigonometric functions at these angles can bedetermined by using the Fundamental identities.

θ (Degrees) θ (Radians) tan θ csc θ sec θ cot θ

30◦ π6

√3

32 2

√3

3

√3

45◦ π4

1√

2√

2 1

60◦ π3

√3 2

√3

32

√3

3

• ex. Find the exact value of each expression. Do not use a calculator.

(a) 2 sin 30◦ + 4 cos 45◦

(b) 3 + cot π4

(c) csc2 60◦ − sec2 π4

1

Section 7.4: Trigonometric Functions of General Angles

• Def: Let θ be an angle in standard position and let (x, y) be any point on theterminal side of θ, except (0, 0). Then the six trigonometric functions can bedefined for any angle θ (not just acute angles) as follows:

sin θ = yr

cos θ = xr

tan θ = yx

csc θ = ry

sec θ = rx

cot θ = xy

where r =√x2 + y2 and none of the denominators is zero. If a denominator

does equal zero, then the trigonometric function of that angle θ is undefined.

• ex. A point on the terminal side of an angle θ is given. Find the value of thesix trigonometric functions of θ.

(a) (6,−8)

(b)(

12,−√

32

)

1

• The value of the six trigonometric functions at the four quadrantal anglesare:

θ (Degrees) θ (Radians) sin θ cos θ tan θ

0◦ 0 0 1 0

90◦ π2

1 0 not defined

180◦ π 0 −1 0

270◦ 3π2

−1 0 not defined

θ (Degrees) θ (Radians) csc θ sec θ cot θ

0◦ 0 not defined 1 not defined

90◦ π2

1 not defined 0

180◦ π not defined −1 not defined

270◦ 3π2

−1 not defined 0

• Def: Two angles in standard position are said to be coterminal if they havethe same terminal side.

• Note: For an angle θ measured in degrees, the angles θ + 360◦n, where nis any integer, are coterminal to θ. For an angle measured in radians, theangles θ + 2πn, where n is any integer, are coterminal to θ. Thus, we havethe following relationships for the values of the six trigonometric functions ofvarious angles:

2

θ (Degrees) θ (Radians)

sin (θ + 360◦n) = sin θ sin (θ + 2πn) = sin θ

cos (θ + 360◦n) = cos θ cos (θ + 2πn) = cos θ

tan (θ + 360◦n) = tan θ tan (θ + 2πn) = tan θ

csc (θ + 360◦n) = csc θ csc (θ + 2πn) = csc θ

sec (θ + 360◦n) = sec θ sec (θ + 2πn) = sec θ

cot (θ + 360◦n) = cot θ cot (θ + 2πn) = cot θ

• ex. Use a coterminal angle to find the exact value of each expression.

(a) cos 480◦

(b) sin (−585◦)

(c) cot 1215◦

• The sign of the six trigonometric functions depends on which quadrant theangle is in.

– Quadrant I: All six trigonometric functions are positive.

– Quadrant II: sin and csc are positive. The rest are negative.

– Quadrant III: tan and cot are positive. The rest are negative.

– Quadrant IV: cos and sec are positive. The rest are negative.

A helpful way to remember this is through the phrase: All Students TakeCalculus. The first letter of each word tells you which of sin, cos, and/ortan is positive in each quadrant. In the first quadrant, All the trigonometricfunctions are positive. In the second quadrant, Sine (and hence its reciprocalcosecant) are positive. In the third quadrant, Tangent (and hence its recip-rocal cotangent) are positive. And in the fourth quadrant, Cosine (and henceits reciprocal secant) are positive.

• ex. If sin θ > 0 and cot θ < 0, name the quadrant in which the angle θ lies.

3

• Def: Let θ denote an angle that lies in one of the for quadrants. The acuteangle formed by the terminal side of θ and the x-axis is called the referenceangle for θ.

• ex. Find the reference angle for each of the following angles:

(a) 135◦

(b) −600◦

(c) 7π4

(d) −5π6

• Reference Angles Theorem: The value of each of the six trigonometric func-tions is the same at any angle as it is at its reference angle, with the possibleexception of a sign difference.

• ex. Use the reference angle to find the exact value of each expression.

(a) sin 135◦

(b) cos−600◦

4

(c) tan 7π4

(d) sec(−5π

6

)

• ex. Find the exact value of each of the remaining trigonometric functions ofθ.

(a) sin θ = −1024

, θ is quadrant IV

(b) cos θ = 35, 3π

2< θ < 2π

(c) cot θ = −169

, sin θ > 0

5

Section 7.5: Unit Circle Approach; Properties of theTrigonometric Functions

• Def: The unit circle is the circle, centered at the origin, of radius 1.

• Let P = (x, y) be a point on the unit circle. Then the radius along thepositive x-axis and the radius touching P form an angle θ. The coordinatesof the point P can be given in terms of trigonometric functions. In particular,sin θ = y and cos θ = x, so the point P = (x, y) can be given by P =(cos θ, sin θ).

• For the point P on the unit circle, we can use x = cos θ and y = sin θ to definethe remaining trigonometric functions. Doing so we get tan θ = y

x, csc θ =

1y, sec θ = 1

x, cot θ = x

y.

• Since the trigonometric functions can be defined by using the unit circle, theyare sometimes called circular functions.

• ex. Find the values of the six trigonometric functions if P =(

12,−√

32

)is a

point on the unit circle.

• If we have a circle of radius r (which is given by the equation x2 + y2 =r2) and a point P = (x, y) on the circle, then the coordinates are givenby (r cos θ, r sin θ). Using these, we can define the remaining trigonometricfunctions on the circle of radius r by tan θ = y

x, csc θ = r

y, sec θ = r

x, cot θ =

xy.

• The domains and ranges of the trigonometric functions are as follows:

1

Function Domain Rangef(θ) = sin θ (−∞,∞) [−1, 1]

f(θ) = cos θ (−∞,∞) [−1, 1]

f(θ) = tan θ R\{π2

+ nπ}, where n is an integer (−∞,∞)

f(θ) = csc θ R\ {πn} , where n is an integer (−∞,−1] ∪ [1,∞)

f(θ) = sec θ R\{π2

+ nπ}, where n is an integer (−∞,−1] ∪ [1,∞)

f(θ) = cot θ R\ {πn} , where n is an integer (−∞,∞)

• Def: Suppose f is a function and θ is in the domain of f . We call f a periodicfunction if there is a positive number p such that θ + p is also in the domainof f and f(θ+ p) = f(θ). If f is periodic, there may be more than one valueof p with this property. The smallest p is called the period of f .

• All six trigonometric functions are periodic functions. sin θ, cos θ, csc θ, sec θall have period 2π and tan θ, cot θ have period π.

• ex. Use the fact that the trigonometric functions are periodic to find theexact value of each expression.

(a) sin 1125◦

(b) sec 8π3

(c) cot 13π4

• All six trigonometric functions have symmetry. sin θ, cscθ, tan θ, cot θ areall odd functions and cos θ, sec θ are even functions. Thus,

sin (−θ) = − sin θ cos (−θ) = cos θ tan (−θ) = − tan θ

csc (−θ) = − csc θ sec (−θ) = sec θ cot (−θ) = − cot θ

2

• ex. Use the even-odd properties to find the exact value of each expression.

(a) cot(−90◦)

(b) sec (−45◦)

(c) cos(−π

3

)

3

Section 7.6: Graphs of the Sine and Cosine Functions

• The sine function, f(x) = sin x, has the following properties:

1. The domain is the set of all real numbers.

2. The range is [−1, 1].

3. It is an odd function so it is symmetric about the origin.

4. It is periodic with period 2π.

5. The x-intercepts occur at all multiples of π; i.e., at x = . . . ,−2π,−π, 0, π, 2π, . . ..The y-intercept is 0.

6. The maximum value is 1 and it occurs at 2π increments of π2; i.e., at

x = . . . ,−3π2, π

2, 5π

2, . . .. The minimum value is −1 and it occurs at 2π

increments of 3π2

; i.e., at x = . . . ,−π2, 3π

2, 7π

2, . . ..

7. The graph looks like

• The cosine function, f(x) = cos x, has the following properties:

1. The domain is the set of all real numbers.

2. The range is [−1, 1].

3. It is an even function so it is symmetric about the y-axis.

4. It is periodic with period 2π.

5. The x-intercepts occur at π increments of π2; i.e., at x = . . . ,−3π

2,−π

2, π

2, 3π

2, 5π

2, . . ..

The y-intercept is 1.

6. The maximum value is 1 and it occurs at all multiples of 2π; i.e., atx = . . . ,−2π, 0, 2π, 4π, . . .. The minimum value is −1 and it occurs at2π increments of π; i.e., at x = . . . ,−π, π, 3π, . . ..

1

7. The graph looks like

• Note: From the graphs of sinx and cosx, we can see that sinx = cos(x− π

2

).

• Def: The amplitude of the sine or cosine graph is the maximum distance ofthe graph from the zeros (before any vertical shifts have been applied to thegraph).

• Recall: The period of a function is the distance it takes for the functions toget back to where it started or to repeat itself.

• Theorem: If k > 0 then the functions y = A sin (kx) and y = A cos (kx) haveamplitude |A| and period, denoted by T , of T = 2π

k.

2

• Def: One period of the graph of y = sin (kx) or y = cos (kx) is called a cycle.

• ex. Determine the amplitude and period of each function then graph thefunction. Show at least two cycles.

(a) y = −2 sin (4x)

(b) y = cos(

12x)− 1

3

(c) y = 12cos (−πx)

• ex. Find an equation for the graph.

4

Section 7.6: Example Answers

• ex. Determine the amplitude and period of each function then graph thefunction. Show at least two cycles.

(a) y = −2 sin (4x)

(b) y = cos(

12x)− 1

1

(c) y = 12cos (−πx)

• ex. Find an equation for the graph.

y = −3 sin(

π3x)

+ 1

2

Section 7.7: Graphs of the Tangent, Cotangent,Cosecant, and Secant Functions

• The tangent function, f(x) = tan x has the following properties:

1. The domain is R\{

π2

+ nπ}.

2. The range is (−∞,∞).

3. It is an odd function, so it is symmetric about the origin.

4. It is periodic with period π.

5. The x-intercepts occur at all multiples of π; i.e., at x = . . . ,−2π,−π, 0, π,2π, . . .. The y-intercept is 0.

6. The vertical asymptotes occur at π increments of π2; i.e., at x = . . . ,−3π

2,

− π2, π

2, 3π

2, . . ..

7. The graph of y = tanx looks like

• The cotangent function, f(x) = cot x has the same properties as the tangentfunction, except that the x-intercepts occur at π increments of π

2; i.e., at x =

. . . ,−3π2,−π

2, π

2, 3π

2, 5π

2, . . . and the vertical asymptotes occur at all multiples

of π; i.e., at x = . . . ,−2π,−π, 0, π, 2π, . . .. The graph of y = cotx looks like

1

• The cosecant function, f(x) = csc x, has the following properties:

1. The domain is R\ {nπ}.2. The range is (−∞,−1] ∪ [1,∞).

3. It is an odd function, so it is symmetric about the origin.

4. It is periodic with period 2π.

5. The is no x-intercept or y-intercept.

6. The vertical asymptotes occur at all multiples of π; i.e., at x = . . . ,−2π,− π, 0, π, 2π, . . ..

7. The graph of y = cscx looks like

2

• The secant function, f(x) = sec x has the following properties:

1. The domain is R\{

π2

+ nπ}.

2. The range is (−∞,−1] ∪ [1,∞).

3. It is an even function, so it is symmetric about the y-axis.

4. It is periodic with period 2π.

5. The is no x-intercept and the y-intercept hits at 1.

6. The vertical asymptotes occur at π increments of π2; i.e., at x = . . . ,−3π

2,

− π2, π

2, 3π

2, . . ..

7. The graph of y = secx looks like

• ex. Graph each function. Be sure to show at least two cycles.

3

1. y = −2 secx

2. y = tan (2x)

4

3. y = 12csc (πx)− 1

5

Section 7.7: Example Answers

• ex. Graph each function. Be sure to show at least two cycles.

1. y = −2 secx

2. y = tan (2x)

1

3. y = 12csc (πx)− 1

2

Section 7.8: Phase Shifts

• In order to graph the functions y = A sin (kx− φ) or y = A cos (kx− φ),rewrite them as y = A sin

[k

(x− φ

k

)]or cos

[k

(x− φ

k

)], respectively. Then

these unctions have an amplitude of |A|, a period of T = 2πk

and a phase shift

(horizontal shift) of φk. If φ < 0 then the phase shift is to the left. If φ > 0

then the phase shift is to the right.

• ex. Find the ampliude, period, and phase shift of each function. Then grapheach function. Show at least two periods.

(a) y = 2 sin (4x− π)

1

(b) y = −3 cos(

12x+ π

2

)

(c) y = − tan (2πx+ π)

2

(d) y = 2 csc(π4x− π

2

)− 1

3

Section 7.8: Example Answers

• ex. Find the amplitude, period, and phase shift of each function. Then grapheach function. Show at least two periods.

(a) y = 2 sin (4x− π)

(b) y = −3 cos(

12x+ π

2

)

1

(c) y = − tan (2πx+ π)

(d) y = 2 csc(

π4x− π

2

)− 1

2