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Precalculus Midterm Review Date: _______________________ Time: __________________ Length of exam: 2 hours Type of questions: Multiple choice (4 choices) Number of questions: 50 Format of exam: 30 questions – no calculator allowed, then 20 questions calculator allowed After the first 30 questions, you will hand in the non-calculator section and take the calculator section. You may not go back to the non-calculator section. In terms of time you should figure on about an hour for the non-calculator section and an hour for the calculator section. (* problems likely to be in the calculator section) Grading of exam: Raw score = # right minus ¼ number wrong and then the scores will be scaled. In this format, it doesn’t pay to guess if you cannot eliminate any choices. Unit 1 – Radian and Degree Measurement 1. Change from degrees to radians to revolutions 2. Draw an angle in standard position and understand positive and negative measurements 3. Identify angles that are co-terminal with another angle * 4. Change an angle from decimal degrees to degrees-minutes-seconds * 5. Use the arc length formula

!

s = r" and find the 3rd variable given two of them a. solve word problems using this formula 6. Understand the difference between linear velocity v and angular velocity

!

" * 7. Use the arc length formula

!

v = r" to find the 3rd variable given two of them a. solve word problems using this formula Unit 2 – the Basic Trig Functions 1. Know the basic trig functions in terms of opposite, adjacent, and hypotenuse as well as x, y, and r 2. Use the Pythagorean theorem to find x, y, or r given two of them 3. Given a point on the terminal side of

!

" , find the six trig functions of

!

" 4. Be able to find the 6 trig functions of the quadrant angles

!

0°,90°,180°,360° 5. Understand the ASTC relationship to find the signs of trig functions in different quadrants 6. Given the value of a trig function in a specific quadrant, find the other trig functions 7. Know the domain/range of the trig functions (values that sin, cos, tan, csc, sec, and cot theta can take on) 8. Be able to find the reference angle for any angle 9. The

!

30°" 60°" 90° and 45°" 45°" 90° 10. Be able to generate trig functions of special (friendly) angles Unit 3 – Right Angle Trigonometry * 1. Use the calculator to find trig functions of angles a. be able to find csc, sec, and cot of angles * 2. Use the calculator to find arc trig (or inverse trig) functions of angles a. be able to find

!

csc"1,sec

"1,cot

"1 of numbers * 3. Solve right triangles a. angle, hypotenuse b. angle, leg c. leg, hypotenuse d. leg, leg


* 4. Solve real-life applications using right angle trig a. surveying problems b. angle of elevation and depression c. bearing and heading(course) - be sure you can go from bearing to heading and vice versa Unit 4 – Graphs of Trig Functions 1. Know the shapes of the sine and cosine curves 2. Know the definition of the following and be able to determine them when looking at a graph a. amplitude b. period c. critical points d. phase shift (horizontal translation) e. vertical shift (vertical translation) f. range 3. Know the role of a, b, c, and d in the equations

!

y = d ± asinb x " c( ) or y = d ± ascosb x " c( ) and how they affect the curve. 4. Determine an equation of a sinusoid given the graph of it. 5. Given critical points of a sinuoid, determine a possible equation * 6. Predict a particle’s position at a certain time given a problem in harmonic motion 7 Recognize the shape of the tan, cot, csc, and sec functions and where their asymptotes occure Unit 5 – Analytic Trigonometry 1. Know the 8 fundamental trigonometric identities 2. Be able to prove identities a. you will not be required to “prove” identities directly but could be asked for example – which of the following expressions is equal to tan x (where you are given 4 choices). 3. Know the sum and difference formulas for sin, cos, and tan a. given a trig function and quadrant of both A and B, find sin, cos or tan of (A + B) or (A – B) 4. Know the double angle formulas for sin, cos, and tan a. given a trig function and quadrant of A, find sin, cos or tan of 2A. 5. Solving trig equations Unit 6 – Solving Oblique Triangles * 1. Use the Law of Sines to solve a triangle in the form AAS or ASA 2. Identify by drawing whether 0, 1, or 2 triangles are possible for SSA * 3. Use the Law of Sines to solve triangle that are solvable using SSA * 4. Use the Law of Cosines to solve triangles that are in the form of SAS and SSS a. know that not all triangles in the form SSS are solvable – they must obey the triangle inequality * 5. Be able to find the area of an oblique triangle a. Heron’s formula

b.

!

1

2absinC or

1

2ac sinB or

1

2bc sinA

* 6. Solving applications using law of Sines and Cosines


Formulas/definitions to know: Unit 1 – Radian and Degree Measurement Co-terminal angles – angles having the same initial and terminal sides Conversion formula for angles:

!

360° = 2" radians =1 revolution Arc length formula:

!

s = r"

Linear velocity:

!

v =linear units

time

Angular velocity:

!

" =angle

time

Linear-angular velocity conversion:

!

v ="r Unit 2 – the Basic Trig Functions The basic trig definitions in terms of opposite, adjacent, hypotenuse:

!

the sine function : sin" =opposite

hypotenuse the cosecant function : csc" =

hypotenuse

opposite

the cosine function : cos" =adjacent

hypotenuse the secant function : sec" =

hypotenuse

adjacent

the tangent function : tan" =opposite

adjacent the cotangent function : cot" =

adjacent

opposite The basic trig definitions in terms of x, y, and r:

!

the sine function : sin x =y

r the cosecant function : csc" =

r

y

the cosine function : cos" =x

r the secant function : sec" =

r

x

the tangent function : tan" =y

x the cotangent function : cot" =

x

y

The Pythagorean theorem ties these variable together : x2+ y

2= z

2

Signs of trig functions in quadrants: A-S-T-C (all – sin – tan – cos)

Quadrant Angle trig functions:

!

0° " Use the point 5,0( ) : x = 5,y = 0,r = 5

90° " Use the point 0,5( ) : x = 0,y = 5,r = 5

180° " Use the point "5,0( ) : x = "5,y = 0,r = 5

360° " Use the point 0,"5( ) : x = 0,y = "5,r = 5


Domain and range of trig functions:

!

Domain : Range :

sin" : all real numbers -1# y #1 or -1,1[ ]cos" : all real numbers -1# y #1 or -1,1[ ]tan" :" $ 90°,:" $ 270° all real numbers or -%,%( )csc" :" $ 0°,:" $180° y # &1 or y '1 or -%,-1( ]( 1,%[ )sec" :" $ 90°,:" $ 270° y # &1 or y '1 or -%,-1( ]( 1,%[ )cot" :" $ 0°,:" $180° all real numbers or -%,-1( ]( 1,%[ )

!

30°" 60°" 90° and 45°" 45°" 90° triangles Special/Friendly angles any multiple of

!

30° and 45° Draw pictures in correct quadrant and be sure to put in signs.

Reference angles for

!

"

!

Quadrant I : "

Quadrant II : 180°#"

Quadrant III : " #180°

Quadrant IV : 360°#"

Unit 3 – Right Angle Trigonometry Angle of elevation and depression Bearing A bearing is always drawn from the nearest north or south line. Heading (course or direction) A heading is always drawn from the north line using this picture:


Unit 4 – Graphs of Trig Functions Graphing methods: Given the curves

!

y = d ± a sin b x " c( ) or y = d ± acosb x " c( )

The amplitude =

!

a

We define the shape of the curve using this chart:

!

a > 0...curve normal

a < 0...curve reversed

" # $

We define vertical change using this chart:

!

amplitude > 1...vertically stretch

amplitude < 1...vertically shrunk

" # $

The period

!

=360°

b degrees( ) or

2"

bradians( )

We define the horizontal stretch in words:

!

period < 360° or 2!( ) ... compressed

period > 360° or 2!( ) ... elongated

"

# $

% $

The phase shift (or horizontal translation) = c.

!

If c > 0, the curve shifts right, if c < 0, the curve shifts left

!

Critical Points = 0, period

4,

period

2,

3period

4, period

True Critical Points = c, period

4+ c,

period

2+ c,

3period

4+c, period + c

The Vertical Translation = d

!

If d > 0...vertical translation is up d units

If d < 0...vertical translation is down d units

" # $

The range of the function is

!

d " a,d + a[ ] To find the equation in the form

!

y = d ± a sin b x " c( ) or y = d ± acosb x " c( ) by looking at the graph:

1) Decide whether it is a sine or cosine curve. If it “starts” at a high point or low point, it is a cosine curve. If it “starts” in the middle, it is a sine curve. You also must determine if the curve is reversed. If so a < 0.

2) Draw the axis of symmetry. That is the value of d. 3) Find the height of the curve above the axis of symmetry. That is a.

4) Find the period by inspection.

!

period =360°

b then b =

360°

period

5) Is there a shift? If shifted right, c > 0. If shifted left, c < 0. 6) Put it all together.

Graphs of the other trig functions:

!

y = tan x

asymptotes at

x = 90°,270°

!

y = cot x

asymptotes at

x = 0°,180°

!

y = csc x

asymptotes at

x = 0°,180°

!

y = sec x

asymptotes at

x = 90°,270°


Unit 5 – Analytic Trigonometry Trig Identities

!

Reciprocal Identities

csc" =1

sin" sec" =

1

cos" cot" =

1

tan"

Quotient Identities

tan" =sin"

cos" cot" =

cos"

sin"

Pythagorean Identities

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

Sum and difference formulas:

!

sin A + B( ) = sin AcosB + cos Asin B cos A + B( ) = cos AcosB " sin Asin B tan A + B( ) =tan A + tan B

1" tan A tan B

sin A" B( ) = sin AcosB " cos Asin B cos A" B( ) = cos AcosB + sin Asin B tan A" B( ) =tan A" tan B

1+ tan A tan B

Double angle formulas:

!

sin2A = 2sin Acos A

cos2A = cos2

A" sin2

A or 2cos2

A"1 or 1" 2sin2

A

tan2A =2 tan A

1" tan2

A

Unit 6 – Solving Oblique Triangles

Law of sines

!

a

sin A=

b

sin B=

c

sinC

Law of cosines

!

a2

= b2

+ c2" 2bc cosA

b2

= a2

+ c2" 2ac cosB

c2

= a2

+ b2" 2abcosC

Area of triangles:

Heron’s formula:

!

Area = s s" a( ) s" b( ) s" c( ) where s =a + b + c

2

!

Area = 1

2base( ) height( )

!

Area = 1

2bc sinA =

1

2ac sinB =

1

2absinC


SSA situations:

precalculus midterm review - d2ct263enury6r.cloudfront.net · a. solve word problems using this...

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