Unit 5 Study Guide

Section 6.4

� Graph the sine and cosine curves.

� Find and use the amplitude and period to graph

the sine or cosine function.

� Find and use the vertical and horizontal

translations to graph the sine or cosine

function.

Section 6.5

� Graph tangent, cotangent, cosecant, and secant

curves.

� Find the asymptotes of each function and use

them to graph the function.

Section 6.6

� Find the exact value using inverse sine, inverse

cosine, or inverse tangent functions.

� State the domain and range of each inverse

function.

� Evaluate composition functions using the

inverse properties.

� Evaluate composition functions using right

triangle trigonometry.

Section 7.1

� Simplify Trigonometric Identities

� State all the reciprocal, quotient, Pythagorean,

co-function, and negative fundamental

identities.

Section 7.2

� Verify Trigonometric Identities

Section 7.3

� Solve Trigonometric Equations

Section 7.4

� State the Sum and Difference Formulas

� Find exact values using sum and difference

formulas

� Rewrite an expression using sum and difference

formulas

� Prove an identity using sum and difference

formulas

� Evaluate sum and difference formulas given two

values and the quadrants they lie in.

Section 7.5

� State the Double Angle, Half-Angle Formulas

� Simplify using Multiple angle formulas

� Evaluate multiple angle formulas

� Find exact values using multiple angle formulas

� Solve trigonometric equations using double

angle formulas and half angle formulas

� Use the power reducing formulas

6.4 Graphs of Sine and Cosine Functions

Basic Sine Curve y= sin x Basic Cosine Curve y=cos x

Amplitude and Period

General equations for sine and cosine:

y = d + asin(bx-c) and y = d + acos(bx-c)

Amplitude represents half the distance between the maximum and minimum values of the function and is given by

amplitude = |a|.

The amplitude streches or shirnks the graph vertically.

If the a is negative, the amplitude is postive, but the graph will be reflected over the x-axis.

1. Graph the following using amplitude.

a) 3cosy x= b) 1

sin2

y x= −

Period

The period stretches or shrinks the graph horizontally and is represented by Period = 2

b

π.

If 0 < b < 1, then the period is greater than 2π and it is a horizontal stretch.

If b > 1, then the period is less than 2π and it is a horizontal shrink.

2. Graph the following using period.

a) cos3

xy

=

b) sin(2 )y x=

Translations of Sine and Cosine Curves

Horizontal Translation

The constant c in the general equations for sine and cosine creates a horizontal translation (shift).

The graph of y = d + asin(bx+c) or y = d + acos(bx+c) completes on cycle from bx – c = 0 to bx – c = 2π . By solving for x,

you can find the interval for one cycle to be 2c c

xb b b

π≤ ≤ + , where c

b is the phase shift and

2

b

πis the period.

3. Sketch the graph.

a) sin( )4

y xπ= + b) cos( )y x π= −

Vertical Translation

The constant d in the general equations for sine and cosine create a vertical translation (shift).

When d > 0, the graph shifts d units upward.

When d < 0, the graph shifts d units downward.

The graph oscillates about the line y = d.

4. Sketch the graph.

a) 5 siny x= − + b) 4 cosy x= +

5. Find the amplitude, the period, the phase shift, and the key points, then graph the function.

a) 3 2sin(3 )4

y xπ= − + −

b)3

sin(2 )2 2

y xπ= − +

6.5 Graphs of Other Trigonometric Functions

Cosecant

cscx = (1/sinx) and the Period is 2π .

Vertical asymptotes where sinx is zero, which occurs at x nπ= .

Graph by first graphing sinx and then take the reciprocals of the y values.

1. Sketch a graph of:

a) cscy x= b) 3csc( )6

y xπ= − − .

Secant

secx = (1/cosx)

Period is 2π and vertical asymptotes where cosx is zero, which occurs at 2

x nπ π= + .

Graph by first graphing cosx and then take the reciprocals of the y values.

2. Sketch a graph of:

a) secy x= b) 1 sec(2 )y x π= + −

The Tangent Function

The graph of the tangent function is symmetric about the origin, it is an odd function.

tanx=(sinx/cosx), which makes tangent undefined for values at which cosx =0. (That is 2

πand

3

2

π.) When the graph is

undefined at a value of x it has a vertical asymptote at that point.

The period of the tangent function is π , so there are other vertical asymptotes when 2

x nπ π= + .

The general equation for tangent is y = atan(bx-c).

Two consecutive asymptotes can be found by solving the equations 2

bx cπ− = − and

2bx c

π− = .

The midpoint of these asymptotes is the x-intercept.

3. Sketch a graph of:

a) tany x= b) 2 tan4

xy

=

The Cotangent Function

The cotangent function also has a period of π .

From the identity cotx=(cosx/sinx), we know that there are vertical asymptotes where sinx is zero, which occurs at

x nπ= .

Two consecutive vertical asymptotes can be found by solving the equations 0bx c− = and bx c π− = .

4. Sketch a graph of:

a) coty x= b) ( )3cot 2y x= − .

6.6 Inverse Trigonometric Functions

Inverse Sine Function

arcsiny x= (1siny x−= ) if and only if sin y x= ,

Where 1 12 2

x and yπ π− ≤ ≤ − ≤ ≤ .

This interval is chosen because it is one-to-one and takes on a full range of values.

Evaluating the Inverse Sine Function

1. Find the exact value, if possible.

a) 1 1sin

2−

b) arcsin(5) c) arcsin (-1)

d)1 1

sin2

− −

e) arcsin3

2

Inverse Cosine Function

arccosy x= (1cosy x−= ) if and only if cos y x= ,

Where 1 1 0x and y π− ≤ ≤ ≤ ≤ .

This interval is chosen because it is one-to-one and takes on a full range of values.

Evaluating the Inverse Cosine Function

2. Find the exact value.

a) 1 3cos

2−

−

b) arccos(1) c) 1 1

cos2

− −

d) 1 1

cos2

−

e) arcos(-3)

Inverse Tangent Function

arctany x= (1tany x−= ) if and only if tan y x= ,

Where 2 2

x and yπ π−∞ ≤ ≤ ∞ − ≤ ≤ .

This interval is chosen because it is one-to-one and takes on a full range of values.

Evaluate the Inverse Tangent Function

3. Find the exact value.

a) 1 3tan

1−

−

b) arctan(1) c) 1tan 3−

d) 1tan (0)−

e) arctan1

3

Inverse Properties of Trigonometric Functions

If 1 12 2

x and yπ π− ≤ ≤ − ≤ ≤ , then sin(arcsinx) = x and arccsin(siny)=y.

If 1 1 0x and y π− ≤ ≤ ≤ ≤ , then cos(arccosx)=x and arccos(cosy)=y.

If x is a real number and 2 2

yπ π− ≤ ≤ , then tan(arctanx)=x and arctan(tany)=y.

The inverse properties do not apply for arbitrary values of x and y. It is only valid within the interval of the domain.

Evaluate Composition Functions Using Inverse Properties

4. Find the exact value.

a) arccos(cos11

6

π− ) b) tan[arctan(14)]

c)1cos(cos 3 )π−

d) 1 3

sin sin4

π−

5. Use right triangles to evaluate the composition of functions.

a) 2

sin arccos5

−

b) 3

tan arcsin5

6. Write an algebraic expression that is equivalent to cos(arctan 3x).

7.1 Using Fundamental Trigonometric Identities

Fundamental Trigonometric Identities

Reciprocal Identities: Quotient Identities:

1csc

sinθ

θ=

1sin

cscθ

θ=

sintan

cos

θθθ

=

1sec

cosθ

θ=

1cos

secθ

θ=

coscot

sin

θθθ

=

1cot

tanθ

θ=

1tan

cotθ

θ=

Negative Identities: Pythagorean Identities:

sin( ) sinθ θ− = − 2 2sin cos 1θ θ+ =

cos( ) cosθ θ− = 2 21 tan secθ θ+ =

tan( ) tanθ θ− = − 2 21 cot cscθ θ+ =

cot( ) cotθ θ− = −

Cofunction Identities:

sin cos(90 )θ θ= ° − cos sin(90 )θ θ= ° −

tan cot(90 )θ θ= ° − cot tan(90 )θ θ= ° −

csc sec(90 )θ θ= ° − sec csc(90 )θ θ= ° −

Trigonometric Identities – A relationship that is true for all values of the variable for which each side of the equation is

defined. (We use them to simplify expressions and prove other identities.)

Simplifying a Trigonometric Expression

1. sin tan sin( )2

A A Aπ+ − 2.

2 2csc (1 cos )x x−

Factoring Trigonometric Expressions (This will later be used with solving.)

3. 3 2cos cos siny y y+ 4. 2c 1cs θ − 5.

25cos 2cos 3θ θ+ −

Adding Trigonometric Expressions

6. sin cos

1 cos sin

θ θθ θ

++

Rewriting a Trigonometric Expression

7. Rewrite 1

1 sin x+ so that it is not in fractional form.

Trigonometric Substitution

8. Use substitution 2 tan , 02

xπθ θ= < < , to write

24 x+ as a trigonometric function of θ .

Rewriting a Logarithm

9. Rewrite ln csc ln tanθ θ+ as a single logarithm and simplify the result.

7.2 Verifying Trigonometric Identities

Guidelines for Verifying Trigonometric Identities

1. Work with one side of the equation at a time. It is often better to work with the more complicated side first.

2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial

denominator.

3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you

want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents.

4. If the preceding guidelines do not help, try converting all terms to sines and cosines.

5. Always try something. Even paths that lead to dead ends provide insights.

Verifying a Trigonometric Identity

1. 2 1 1

2sec1 sin 1 sin

xx x

= +− +

2.2 2 2(tan 1)(cos 1) tanx x x+ − = −

3.cos

sec tan1 sin

xx x

x+ =

− 4.

4 2 2 2tan tan sec tanx x x x= −

7.3 Solving Trigonometric Equation

The goal of solving a trigonometric equation is to get the trigonometric function alone by using algebraic methods that

we use to get x alone in an equation. Once the trigonometric function is alone we can use the inverse trigonometric

functions to solve for the angle measure.

All trigonometric functions are periodic and so there will be infinitely many solutions to the equation. Another solution

to the equation can be found by adding the period of the function to your solutions that lie within this period.

Answers should be written in radians in terms of π .

Solve for the missing angle.

1. tan 2 tanx x− = − 2. 24 cos 3 0x − =

3. sec sin 2sinx x x= 4. 2cos 3sin 3x x− =

5. 22sec tan 5x x+ = 6. 2 cos 1 0

2

θ − =

7. ( )2tan 2 3x =

7.4 Sum and Difference Formulas

Sum and Difference Formulas

sin( ) sin cos cos sin

sin( ) sin cos cos sin

cos( ) cos cos sin sin

cos( ) cos cos sin sin

tan tantan( )

1 tan tantan tan

tan( )1 tan tan

u v u v u v

u v u v u v

u v u v u v

u v u v u v

u vu v

u vu v

u vu v

+ = +− = −+ = −− = +

++ =−

−− =+

Evaluating a Trigonometric Function

1. Evaluate cos 285° .

2. Find the exact value of 5

sin12

π.

3. Prove that the given equation is an identity.

sin cos2

x xπ − =

.

4. Simplify. tan( 3 )θ π+

5. Find all solutions of cos( ) cos( ) 1, 0 24 4

x x xπ π π+ + − = − ≤ <

6. Suppose that 3

sin5

u = and 15

sin17

v = , where 02

u vπ π< < < < . Find sin( )u v+ and cos( )u v− .

7.Verify that sin( ) sin sinh 1 cosh

(cos ) (sin ) , 0x h x

x x where hh h h

+ − − = − ≠

8.Write cos(arctan1 arccos )x+ as an algebraic expression.

7.5 Multiple Angle Formula

Double Angle Formulas Half Angle Formulas

2 2

sin 2 2sin cos

cos2 cos sin

u u u

u u u

=

= −

1 cossin

2 2

u u−= ±

21 2sin u= − 1 cos

cos2 2

u u+= ±

22cos 1u= −

1 cos

tan2 1 cos

u u

u

−= ±+

sin

1 cos

u

u=

+

1 cos

sin

u

u

−=

Power Reducing Formulas

2

2

2

1 cos 2sin

21 cos 2

cos2

1 cos 2tan

1 cos 2

uu

uu

uu

u

−=

+=

−=+

1. Use the double angle formula to rewrite the equation.

26cos 3y x= −

22 tan

tan 21 tan

uu

u=

−

2. If 3

sin ,5 2

u uπ π= < < , find sin 2 ,cos 2 , tan 2 ,sin

2

uu u u

3. Derive the triple angle formula. sin3x

Solving Trigonometric Equations

4. 2cos sin 2 0, 0 2x x x π+ = ≤ < 5. cos2 1 sinx x= − for 0 2x π≤ < .

6. 3cos2 cos 2x x+ = for 0 2x π≤ < . 7. 2 22 2sin 2cos ,0 2

2

xx x π− = ≤ <

Power Reducing Formulas

8. Rewrite 4sin x as a sum of first powers of the cosines of multiple angles.

Trig Identities Practice

Simplify.

1. 1 1

sec tan sec tant t t t−

− + 2.

2 2 2cos cos siny y y+

3. sec cot cot cosx x x x− 4.

2

2 2

cos 1

cos tan

x

x x

−

Verify.

5. ( )( ) 1cotcsccotcsc =−+ θθθθ 6. csc cotsin

cosx x

x

x+ =

−1

7. sin cot cos cscx x x x+ = 8.

22

2

sec 1sin

sec

xx

x

− =

Things to Memorize for This Exam

Basic Sine Curve y= sinx (In Degrees, radians, or decimal radians)

Basic Cosine Curve y=cosx (In Degrees, radians, or decimal radians)

� amplitude = |a| Period = 2

b

π.

� To find Phase Shift for sine and cosine: bx – c = 0 to bx – c =2π

� To find two consecutive vertical asymptotes for tangent: 2

bx cπ− = − and

2bx c

π− =

� To find two consecutive vertical asymptotes for cotangent: 0bx c− = and bx c π− =

� For cosecant graphs; vertical asymptotes are where sinx is zero.

� For secant graphs; vertical asymptotes are where cosx is zero.

Inverse Sine Function

arcsiny x= (1siny x−= ) if and only if sin y x= ,

Where 1 12 2

x and yπ π− ≤ ≤ − ≤ ≤ .

Inverse Cosine Function

arccosy x= (1cosy x−= ) if and only if cos y x= ,

Where 1 1 0x and y π− ≤ ≤ ≤ ≤ .

Inverse Tangent Function

arctany x= (1tany x−= ) if and only if tan y x= ,

Where 2 2

x and yπ π−∞ ≤ ≤ ∞ − ≤ ≤ .

Inverse Properties of Trigonometric Functions

If 1 12 2

x and yπ π− ≤ ≤ − ≤ ≤ , then sin(arcsinx) = x and arccsin(siny)=y.

If 1 1 0x and y π− ≤ ≤ ≤ ≤ , then cos(arccosx)=x and arccos(cosy)=y.

If x is a real number and 2 2

yπ π− ≤ ≤ , then tan(arctanx)=x and arctan(tany)=y.

Fundamental Trigonometric Identities

Reciprocal Identities: Quotient Identities:

1csc

sinθ

θ=

1sin

cscθ

θ=

sintan

cos

θθθ

=

1sec

cosθ

θ=

1cos

secθ

θ=

coscot

sin

θθθ

=

1cot

tanθ

θ=

1tan

cotθ

θ=

Pythagorean Identities: Co-function Identities:

2 2sin cos 1θ θ+ = sin cos(90 )θ θ= ° − cos sin(90 )θ θ= ° −

2 21 tan secθ θ+ = tan cot(90 )θ θ= ° − cot tan(90 )θ θ= ° −

2 21 cot cscθ θ+ = csc sec(90 )θ θ= ° − sec csc(90 )θ θ= ° −

Sum and Difference Formulas

sin( ) sin cos cos sin

sin( ) sin cos cos sin

cos( ) cos cos sin sin

cos( ) cos cos sin sin

tan tantan( )

1 tan tantan tan

tan( )1 tan tan

u v u v u v

u v u v u v

u v u v u v

u v u v u v

u vu v

u vu v

u vu v

+ = +− = −+ = −− = +

++ =−

−− =+

Double Angle Formulas Half Angle Formulas

2 2

sin 2 2sin cos

cos2 cos sin

u u u

u u u

=

= −

1 cossin

2 2

u u−= ±

21 2sin u= − 1 cos

cos2 2

u u+= ±

22cos 1u= −

*You also need to have memorized everything that we

needed to memorize for previous exams!