Precalculus Spring 2017

Exam 5 Summary (Section 6.6 through 8.2)

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 6.7

� Solve a right triangle using right triangle trigonometry, the sum of the angles of a triangle, and the Pythagorean

Theorem.

� Draw a navigational bearing.

� Solve a right triangle problem that uses navigation.

� Find the frequency of a simple harmonic motion.

Section 8.1

� Find the missing sides and angles of a triangle using the Law of Sines

� Find the area of a triangle using 1

sin2

A ab C=

Section 8.2

� Find the missing sides and the cosine of the angles using the Law of Cosines

� Find the area of a triangle using Heron’s Area Formula

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

6.6 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 1

sin2

−

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 3

cos2

− −

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 3

tan1

− −

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).

6.7 Applications and Models

Right Triangle Applications

Solve the right triangle. (Find all the missing sides and angles using right triangle trigonometry)

*Remember lower case letters represent side lengths and upper case letters represent angle measures.

1. A

c b=14

B a C = 32.9°

2. A safety regulation states that the maximum safe rescue height for a fire department’s ladder when at an angle

of elevation of 82° is 115 feet. How long is the ladder?

3. The wave pool (Tsunami Bay) at Magic Waters water park in Rockford, IL is 100 feet long and 60 feet wide. The

bottom of the pond is slanted so the depth is 6 inches at the shallow end and 5 feet at the deep end. Find the

angle of depression of the bottom of the pool.

Trigonometry and Bearings

- North is a bearing of 0 degrees.

- Compass bearings and courses are given with three digits traveling clockwise from north. (Ex. 070° or 112° )

4. Draw a earing

of 012° .

5. Draw a bearing

of 280° .

6. Draw a bearing

of S47° E.

7. Draw a bearing

of N 31° W.

8. A ship leaves port at noon and has a bearing of S 29° W. The ship sails at 20 knots. How many nautical miles

west will the ship have traveled by 6:00 P.M.?

9. A ship leaves port at noon on a course of 035° at a speed of 20 knots. At 2PM the ship changes course to 305° at

a speed of 25 knots. Find the ships distance from the port of departure at 3PM when it is due north of its

starting point.

Harmonic Motion

A point that moves on a coordinate line is said to be in simple harmonic motion if its distance d from the origin at time t

is given by either sind a tω= or cosd a tω= , where a and ω (the angle) are real numbers such that 0ω > . The

motion has amplitude |a|, period 2πω

, and frequency 2

ωπ

.

10. Write the equation for the simple harmonic motion of a ball attached to the bottom of a spring. Suppose that

10 centimeters is the maximum distance the ball moves vertically upward or downward from its equilibrium

position and that the time it takes to move from its maximum displacement above zero to its maximum

placement below zero is 4 seconds.

11. Given the equation for simple harmonic motion 1

sin164

d tπ= , find:

a) The maximum displacement.

b) The frequency.

c) The value of d when t = 3.

8.1 Law of Sines

Oblique Triangles: Triangles with no right angles.

Law of Sines

Use to solve AAS, ASA, or SSA (ambiguous case) oblique triangles.

If ABC is a triangle with sides a, b, and c, then

sin sin sinA B C

a b c= = or the reciprocal form

sin sin sin

a b c

A B C= =

Solving a Triangle Using the Law of Sines

1. Solve ABC∆ if <A = 45° , <B = 30° , a = 14.

2. Solve ABC∆ if <B = 84° , a = 18, b = 9.

3. Solve ABC∆ if <B = 120° , b = 15, c = 11.

4. Solve QRS∆ if <S =70° , r = 26, s = 25.

Area of an Oblique Triangle

K = Area of

K =

K =

K =

K =

By right triangle trigonometry we know , so h = a sinC

5. Two sides of a triangle have lengths 12 cm and 8 cm. The angle between them is 60° . Find the area of the

triangle.

6. Find the area of a triangular lot having two sides of lengths 9 meters and 15 meters and an included angle of

150°.

ABC∆1

2bh

1sin

2ab C

1sin

2bc A

1sin

2ac B

sinh

Ca

=

8.2 Law of Cosines

The law of cosines can be used to find missing information of a triangle given SAS or SSS.

Finding the largest angle first is helpful because we can determine if using the Law of Sines would make solving for the

other angles easier.

2 2 2 2 cosc a b ab C= + −

1. A triangle has side lengths 6, 9, 14. Find the cosine

of all the angles.

2. The lengths of a triangle are 10 cm and 6 cm. The

angle between them is 120° . Find the remaining side

and the cosine of the remaining angles.

Heron’s Area Formula

( )( )( )

( ) / 2

Area s s a s b s c

s a b c

= − − −

= + +where,

3. Find the area of a triangle with side lengths 3,

5, and 4 meters.

4. Find the area of a triangle with side lengths 5,

6, and 7 feet.

7.1 Using Fundamental Trigonometric Identities

Fundamental Trigonometric Identities

Reciprocal Identities: Quotient Identities:

1csc

sinθ

θ=

1sin

cscθ

θ=

sintan

cos

θθ

θ=

1sec

cosθ

θ=

1cos

secθ

θ=

coscot

sin

θθ

θ=

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 − 5.25 cos 2 cos 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

xπ

θ 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. 22 sec 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 tan

tan tantan( )

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 v

u vu v

u 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

2

1 cos 2cos

2

1 cos 2tan

1 cos 2

uu

uu

uu

u

−=

+=

−=

+

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

26cos 3y x= −

2

2 tantan 2

1 tan

uu

u=

−

2. If 3

sin ,5 2

u uπ

π= < < , find sin 2 ,cos 2 , tan 2 ,sin2

uu u u

3. Derive the triple angle formula. sin 3x

Solving Trigonometric Equations

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

6. 3cos 2 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 2

bx cπ

− =

� 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

θθ

θ=

1sec

cosθ

θ=

1cos

secθ

θ=

coscot

sin

θθ

θ=

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 tan

tan tantan( )

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 v

u vu v

u 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!