Praxis 2 Trig

8/10/2019 Praxis 2 Trig

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MATHEMATICS:CONTENTKNOWLEDGE (0061)TRIG TEST 1(PRAXIS2MATH.COM)________________________________________________________________________

READ THE FOLLOWING BEFORE YOU BEGIN!

You are about to embark on a trigonometric journey.

eah thats a real word, Google it.) Those reading this probably

avent had a real Trig coursein: 6 years for some, 7+ years for

any, or in the case of myself, never at all. High school didnt

fer it, colleges combined it with other courses, and when all wasid and done, some of us only had a vague idea of what

igonometry really was. The following exam is going to test your

omprehension on Trigonometry. What makes me qualified to test

ou? The high school that hired me gave me Trig as my first

ourse. I have successfully taught dozens of students (nearing

undreds now), and as a result, our schools NECAP scores have

mproved in that area. Enrollment in Trig has also increased during

e time I have been teaching at my high school. According to one

my students: You, like, totally rock and stuff!

So before you begin this journey, I will give you a crash-

ourse in Trigonometry. Consider the following two pages your

iniature lesson in Trig. I will gladly answer questions via email if

ou ask. Half of the time I will be able to respond to you within

inutes (unless Im in school or sleeping, all Eastern time), so dont

esitate to ask. Lets begin

Trigonometry, loosely translated, means the study of

angles. Simplified even more: its the relationship between the

des of a triangle. Thats it. Very simple, but theres so much

ore than that.

The basic triangle is the right triangle, which is obviously a

angle that has a right angle (90 degrees). The side opposite theght angle is called the hypotenuse, which is always the longest

de of a right triangle. Right triangles can easily vary in sizes, so

e easiest right triangle for us to study is a triangle which has a

ypotenuse of 1. Now if you were to draw a few right triangles on

n xy-plane using a hypotenuse of 1, it would look something like:

Each triangle in the

picture has a hypotenuse

of 1, and a right angle

that sits on the x-axis.

The angle formed by the

hypotenuse and the x-axis is called the central

angle. Its a pretty

important angle for

beginners in

Trigonometry. Now , if

we were to connect the

end-points of all the

hypotenuses (hmm, the

plural version sounds

nny to say), we would get something like this:

Do you notice somet

We ended up drawin

circle around the orig

the xy-axis. This circ

called the unit circle

were to draw any lin

starts at coordinate (and draw it out towa

the edge of this unit

(a radius), its length

would be 1 unit. Fro

that line, I could draw

another line straight

to the x-axis, and usin

that x-axis, we would

a triangle. I could dr

infinite number of triangles with a hypotenuse of one, just by

the unit circle as a guide.

So heres where the real trig comes into play. Using

unit circle as a guide, if we were to draw a triangle with a cent

angle of 30o, it would look like something below:

= 30o)

Now we can study t

relationship of side

using this triangle a

as our guide. Lets

begin:

The ratio of the side

opposite divided the hypotenuse can

written as: ( ). Th

called the sin of

(sin ). If you were

put sin(30) in your

calculator, you will

an answer of . So: sin = , and since our hypotenu

1, our opposite side (y) must be equal to one-half.

The ratio of the side adjacent to divided by the hypotenuse

be written as: ( ). This is called the cos of (cos ). If you w

to put cos(30) in your calculator, you will get an answer of

cos = , and since our hypotenuse is 1, our adjacent

(x) must be equal to .

The ratio of the side opposite to divided by side adjacent to

can be written as: ( ). This is called the tan of (tan ). If yo

were to put tan(30) in your calculator, you will get an answer

So: tan = , written properly is: .


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K so lets recap:

n =

os =

n =

ver hear of a little acronym pronounced so-cuh-toe-uh? Well it

oks like this: SOHCAHTOA. Look at it carefully, and then look at

e recap I wrote just above it. SOH (Sin = O / H), CAH (Cos = A / H),

OA (Tan = O / A). Just remembering SOHCAHTOA will help you

th a few problems on the 0061 exam itself.

nd if you havent noticed yet, weve only dealt with 3 ratios of

des, there are still 3 more to go. Ill take it one step at a time:

n = If we are to take the reciprocal of this, we get .he reciprocal is called csc (cosecant). So: csc =

os = If we are to take the reciprocal of this, we get .

he reciprocal is called sec (secant). So: sec =

n = If we are to take the reciprocal of this, we get .

he reciprocal is called cot (cotangent). So: cot =

heres no handy-dandy little acronym or anything for the other

ree trig functions, they are just the reciprocals of the first three.

should also be noted that a triangle can exist in any of the 4

uadrants on an xy-plane, thus resulting in negative fractions. The

ypotenuse is never negative, but the adjacent and opposite sides

a triangle can be. Heres a little chart explaining what Im talking

bout below. Take a big look before you continue:

You should have noticed a little helpful reminder in t

diagram for you. All Students Take Calculus, which means A

trig functions are positive in the first quadrant, only Sin is pos

in the second, only Tan is positive in the third, and only Cos is

positive in the fourth. The reciprocals work the same way: A

positive in the first, csc is positive in the second, cot is positive

the third, sec is positive in the fourth.

So thats the basics of Trig. Im not going to go any

farther, because much of what you need to know is explained

the answers of the following questions. Im also not going to

into explaining the functions part of this exam before you tak

If youve taken my other practice exams, the functions part of

exam should be easy for you. Other components of trigonom

are shown below:

= =

a2= b

2+ c

2(2 b c cos A)

Cos A =

In a right triangle: a2+ b

2= c

2

It should be noted that in triangles, capital letters (A, B, C, etc

denoted as angles, while lower case letters (a, b, c, etc) are th

sides opposite those angles. Dont screw that up. Labeling a

triangle incorrectly is an automatic wrong answer on your exa

In an almost entirely unrelated note, cos spelled correctly is

cosine, and sin is spelled sine. We use cos and sin for sho

just want to clear this up now in case there is confusion later o

whatever reason. You most likely will see sin, cos, tan, etc on

0061 exam, but who knows really.

You should treat this exam just like the other ones you purcha

from me (assuming you did). Give yourself a calculator ready

go. Rid yourself of any and all distractions for a total of 55

minutes. Pencils and scrap paper should be ready to go. Mak

sure you do not cheat, do not pause, nothing. Treat it like the

actual exam. Dont forget to use your equations sheet. Good

(PSThe following questions are 0061 quality)


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) There exists a right triangle PQR (not shown), where angle

PQR is a right angle, angle RPQ is 29o, and side PR is 12 units.

Find the length of side RQ.

(A) 10.50 units

(B)

13.72 units(C) 24.75 units

(D) Not enough information provided.

) There exists a triangle on an xy-plane that has a central angle

. If sin = , and tan in which quadrant is the

triangle located?

(A) Quadrant I

(B)

Quadrant II

(C) Quadrant III

(D) Quadrant IV

) A kid accidentally threw his Frisbee on top of a building that

has a flat roof. He needs to get a ladder, but first he needs to

find out how tall the building is. He is 15 feet away from the

building, and the angle of elevation to the top of the building

is 58o. Which answer below best represents the height of the

building?

(A) 8 feet

(B) 13 feet

(C) 24 feet

(D) 36 feet

) A summer camp is constructing a zip-line that will hang from

a 100-foot pole. The end of the line will be fastened to a

platform that is 5 feet off the ground. If the angle of

elevation from the platform to the top of the pole is 25o, then

what is the amount of wire they will need in order to make

the zip-line.

(A) 224.79 feet

(B) 236.62 feet

(C) 248.45 feet


OK so heres the deal. If you found the first four questions

to be troublesome, then you should skip to the answer

sheet NOW and see if you got the first four right. There is

no point in continuing with this exam if you have no clue

what youre doing, correct? Go to the answer sheet, see

what the answers are, read the explanations, and make sure

you know what you are doing right and what you are doingwrong. Only when you fully understand your mistakes and

feel comfortable with yourself should you continue.

But if you feel confident so far, then by all means go on to

number 5.

5) A car is travelling up a straight road that has an

inclination of 13o. If the car initially started at sea level,

then how high above sea level is the car after it has

travelled for 6 miles?

(A) 1.35 miles

(B) 1.36 miles

(C) 1.37 miles

(D) 1.39 miles

6) A woman is skiing down a mountain with a vertical heigh

1750 feet. The distance from the top of the mountain to

bottom is 3200 feet. What is the angle of elevation?

(A) 28.67o

(B) 33.15o

(C) 56.85o

(D) 61.32o

7) A man was standing 9 feet away from a sign that was res

on top of a post. He realized that the angle of elevation

where he stood to the bottom of the sign was 29o, and t

angle of elevation to the top of the sign was 40o. How ta

the sign?

(A) 1.75 ft

(B) 2.03 ft

(C) 2.27 ft

(D) 2.56 ft


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) There exists triangle JKL (not shown) that has side lengths of

21, 31, and 35. Which answer below best represents the

biggest angle in triangle JKL?

(A) 72o

(B)

76

o

(C) 82

o

(D) 89o

) There exists a utility pole that normally sits at a 90 degree

angle with the ground. However, a wind storm has pushed

the pole over an extra 8odirectly towards the sun. The

shadow (which is directly away from the sun) yields a

measurement of 17 feet, and the angle from the end of the

shadow to the top of the pole is 48o. Which answer best

represents the height of the pole?

(A) 13 feet

(B) 17 feet

(C) 18 feet

(D) 23 feet

0) Using information given in the following diagram, which

answer best represents the measurement of side a?

(A) 9 cm

(B) 11 cm

(C) 13 cm


11) Me and a friend of mine met for coffee at a small town i

Vermont (where I live, true story). When we departed, h

drove at a 37oangle above route 7 in a straight line head

north east. I drove 23obelow route 7 in a straight line

headed south east. If he drove a total distance of 21 mil

and I drove for 16 miles, which answer below best repre

how far away we were from each other when we reachedestinations?

(A) 18 miles

(B) 19 miles

(C) 24 miles

(D) 26.5 miles

12)

The sun shined down on a man who was 73 inches tall, wproduced a shadow that was 102 inches long on the grou

What is the angle of elevation of the sun?

(A) 35.59o

(B) 45.70o

(C) 54.41o



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3) An airplane takes off from an airport flying in a straight line at

a steady rate of 400 miles per hour, with an angle of elevation

of 8o. After 2 minutes of flight, how high off the ground is the

airplane?

(A) 0.98 miles

(B) 1.86 miles

(C) 55.67 miles

(D) 95.80 miles

4) A pilot is sitting in an airplane that is 6,000 feet in the air. He

looked out his window and saw Town A at an angle of

depression of 29o. Out of the same window, he saw Town

Bat an angle of depression of 67o. Which answer below

best represents how far apart the two towns were from eachother?

(A) 4,688 feet

(B) 7,680 feet

(C) 8,277 feet

(D) 10,809 feet

he next problem is the MOAP (Mother Of All Problems). It

ont be easy and I dont expect anybody but the best to get itght. So if you think you can conquer this beauty, then go ahead

nd continue. Otherwise, check your first 14 answers, see if they

e right and if you know what you are doing. If you are all set

ter that, then continue:

15) The diagram below is of scalene triangle ABC. This triang

has a line BD that bisects AC. Angle ABC = 58o, while AC

cm, and BC = 12 cm. Using the values given, what is the

approximate value of BD?

(A)

8.5 cm

(B) 10.5 cm

(C) 14 cm

(D) 15 cm

This is the end of the test. I do not have any questions that swhat is the minimum or maximum of such-and-such trig

equation. You could easily graph any such question on you

calculator using your y= function, and simply zoom in at the

answer. For example:

What is the MAX of: y = 3 sin (x)4?

If you go to y= on your calculator, put in the equation above

hit zoom and 7 (Trig), it will graph the curve for you, and

you can see the MAX would be -1. Anytime you are asked to

the max and/or min of a sin/cos curve, just graph it and look

too easy.


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DETAILED ANSWERS

1) ANSWER: (B) 13.72 units

This is your basic trigonometric question:

The first thing we should do is draw this thing up so we know

what it looks like. Make sure you label everything correctly.

Just to make it a little simpler, I also labeled the side we are

looking for with an x:

Alright here we go. What we first need to do is figure out

which trig function we are going to use to find x. If we are to

use the angle RPQ, then we are dealing with the sides which

are adjacent and hypotenuse to our angle. SOHCAHTOA, tells

us to use Cosine.

cos =

Now fill it in:

cos 29 =

If you cross multiply and divide correctly, you will see that:

x =

Put this in your calculator and you will get:

x = 13.72 units, choice (B)

2)

ANSWER: (B) Quadrant II

This kind of question can be difficult unless you know

the process. Lets look at what we know:

I.

sin = , and tan . Since sin is equal to theopposite side divided by the hypotenuse, and that the

hypotenuse can NEVER be negative, we know that our

opposite side must be positive (because our fraction is

positive). The opposite side is the side that goes up and

down (see page 1), which is the y value. So if the y value

must be positive, then the triangle must be in either

quadrants I or II.

II. If tan is negative (tan , so it must be negative),

then either the opposite or adjacent sides must be

negative. Since we know the opposite side is positive,

then the adjacent side must be negative. The onlyquadrant where the opposite side is positive and

adjacent is negative is in Quadrant II, choice (B).

For more clarification, check out the work I did for you

on page 2. But if you are lazy, heres the picture I gave

you:


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3)

ANSWER: (C) 24 feet

First step, draw a picture. Noticing a pattern are we?

Here is the picture that I drew:

Next, we have to know what trig function to use. Well, we

know the value of the side adjacent to the angle, and we

need to figure out the value of the side opposite to the angle.

Going over our acronym (SOHCAHTOA) helps us figure out

that we need to use the tangent function.

Tan 58 =

Which can be re-written as:

15 Tan 58 = x

And if you put that in your calculator, you will get 24.005,

which is closest to the answer (C), 24 feet.

4)

ANSWER: (A) 224.79 feet

Guess what my first step is:

I. Picture:

II. See what I did there? This question is a little different

than the first three. Yes, the pole is 100 feet high, but

the triangle we are using as a height of 95 feet because

of the platform down below.

III. Next, we need to figure out which trig function to use.

We are dealing with the side opposite our angle, and the

hypotenuse. Using SOHCAHTOA, we see that we needto use sin.

IV. Sin 25 = Cross multiply

95 = (Sin 25)(x) Divide by Sin 25

= x

x = 224.79 feet, choice (A)


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5)

ANSWER: (A) 1.35 miles

Gee, I wonder what Ill do to start this one diagram? Yes

please.

Here we go:

Just like the other questions before this one, we need to

figure out which trig function to use. Well, were looking at

the side opposite the angle, and the hypotenuse. This meanswere using sin.

sin 12 = Re-write it

6 sin 12 = x, so x = 1.35 miles, choice (A).

6)

ANSWER: (B) 33.15o

Finally we have a problem in which we have to determin

angle, not a side measurement:

I. Take a stab at what Im doing first.. yup:

II. Once again, we need to first determine which trig functi

are to use. Well, I see a side opposite angle x, and we

know the hypotenuse. This tells me to use the sin functi

III. Set it up: sin x = Find the inverse of sin to solve

The inverse of sin is called arcsin, and is found on most

Calculators by doing:

2nd

(blue button), sin. This will show you sin-1

. So do:

sin-1

(1750/3200)Dont forget to end your parenthes

wont screw up this problem, but its a good habit to get

Your calculator will show you 33.15o, choice (B)


9/14


7)

ANSWER: (D) 2.56 ft

Ahh, the first real tough trig problem. However, this is a

relatively simple problem disguised as a tough one. Did you

make it harder than it has to be? Lets take a look:

First, you have to make sure you know whats being asked.

The question asks how tall the sign is, not how tall off the

ground it is or how tall the sign & post are, just the sign. The

second thing you should know is angle of elevation. Angle

of elevation (also known as angle of depression, well get to

that later) is the measurement from the bottom to the top of

something. In this case, its the measurement from the

ground to the bottom and top of the sign. I would draw a

picture next.

The picture(s) should look something like this:

As you can see, the angles technically make two triangles. Ifwe can somehow find the height of the shorter triangle (x)

and subtract it from the height of the taller triangle (y), we

should be able to find out how tall the sign is.

We have two parts of the triangle, an angle and a side. Thats

a good start, so now we need to decide which trig function to

use in order to find x and y. Well, we have the side opposite

the angle, and the side adjacent to the angle.

Opposite/Adjacent = Tangent (TOA). Now we start solving.

V. Since were using tangent, we get the following

equations:

tan 40 = which can be re-written: y = 9 tan (40)

tan 29 = which can be re-written: x = 9 tan (29)

Simplified, we get: y = 7.55 ft

x = 4.99 ft

Subtract them from each other and we get 2.56 ft,

choice (D).


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8)

ANSWER: (C) 82o

Looks like a lot of work to figure out the answer, but it should

actually take one step:

We are given a triangle, and the measurement of all three

sides of that triangle. Your first instinct might be to find all

three angles, and then pick the biggest angle of the three as

your answer. However, your biggest angle is always the angle

opposite the biggest side. Our biggest side in this triangle is

35 units.

We could draw a picture if we wanted, or in this case, we

could jump straight to the equation since we already know

what we want to find. We want to use an equation that uses

all three sides, and finds an angle. This is the best one to use:

Cos A =

If were finding angle A, then side a must be 35 units. The

other two (b and c) are 21 and 31 units. It doesnt matter

which is which, so we can label it accordingly:

Cos A =

Cos A =

Cos A = .136 Take the arcos of both sides and you get:

A = 82.19o, which is closest to choice (C).

9)

ANSWER: (D) 23 feet

This question isnt as obvious as some of the others.

I. The first thing I will do this time is something completely

and totally different draw a picture:

II. Do you notice something about the diagram that isnt

mentioned in the problem? I labeled the top angle 34o.

I know its 34 because 180 4898 = 34. With that, I

can now use the law of sines to figure out the answer to

x.

III. = Cross multiply.

17 (sin 48) = x (sin 34) Divide both sides by (sin 34).

= x Do it out.

22.59 = x Which is closest to 23 feet, choice (D).


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0)

ANSWER: (C) 13 cm

Im hoping to maybe catch a few people off-guard with this

question.

You should have looked at the diagram, tried to figure out

what to do, and then realized something

This triangle has 2 angle measurements, so in order to get the

third, we could just simply subtract the other two

measurements from 180. In doing so, we know that the top

angle is 76o. We cant figure out the answer without it.

Knowing all angle measurements, we can now use the law of

sines:

= Cross multiply.

13.5 (sin 65) = x (sin 76) Divide both sides by (sin 34).

= x Do it out.

12.61 = x Which is closest to 13 feet, choice (C).

11)

ANSWER: (B) 19 miles

Did you find this to be tricky? Some have, but if you hav

good for you!

I. Lets take a look at the diagram:

II. What we really have here is one big triangle disguised as

perhaps two triangles. What we want to find out is the

distance between two points. We know two sides of a

triangle, and an angle that is opposite of the third side

(37o+ 23

o= 60

o). Time to set this up:

III. c2= 16

2+ 21

2(2 16 21 cos 60) One step at a tim

c2= 697(336)

c2= 361 Take the square root of both sides.

c = 19 miles, choice (B).


12/14


2)

ANSWER: (A) 35.59o

All the information you needed is right there.

Lets take a look at the diagram:

Looks like we have ourselves a right triangle. The angle we

want to find (x) is shown on the diagram. We know the

measurements of the side opposite (73 inches) and adjacent

(102 inches) to the angle. Opposite / Adjacent = Tangent.

Tan x = Take the arctan of both sides (Press: 2nd

, tan)

x = 35.59o, choice (A).

13)

ANSWER: (B) 1.86 miles

Did you do your conversions first?

I. Draw it:

Dont make fun of my airplane...

II. We arent given the exact bits of information we need, b

we can use what we have to get our y value as shown

picture. Since the plane is travelling at 400 miles per ho

its we can take 400 and divide it by 60 minutes to see ho

fast its going per minute.

400 / 60 = 6.6666 miles per minute Multiply it by 2, s

the airplane flew for 2 minutes.

The plane has travelled 13.33 miles!

III. Since our y value = 13.33333, we can now setup our equ

We know our angle (8o), and our hypotenuse. Since we

to find out our opposite side, we must use sin to f igure it

sin 8 = Multiply both sides by 13.333.

13.333 sin 8 = x Simplify

x = 1.86 miles, choice (B).


13/14


4)

ANSWER: (C) 8,277 feet

Fun to do!

Draw it again:

The above drawing will make you go cross-eyed if you look at

it long enough. Angle of Depression means the angle going

downward from the horizontal line from the height of the

plane. Actually, angle of depression = angle of elevation most

of the time, but this Praxis II 0061 exam uses confusing

wording, and I dont trust it one bit.

Look closely at our drawing, and you should notice that

there are two triangles. Both triangles share the same

side (6,000 feet), so Ill just redraw those two triangles

separately:

IV. Now we need to determine which trig function to use.

We know the side opposite the angle, and want to find

the side adjacent to the angle. Opposite / Adjacent =

tangent, so:

V.

Tan 67 = and: Tan 29 =

Now heres a little hint for you; when you have a

fraction equal to a fraction, you can legally switch the

diagonal values, and it wont change the fact that they

are still equal to each other. Watch:

= This is true. Now switch the 3 and the 8.

= Still true! Now switch the 4 and the 6.

= Still true! This will save you time on the exam

So anyway:

Tan 67 = and: Tan 29 =

Tan 67 is like saying its over one, so like I did above,

lets switch the diagonals by switching the Tan 67 with x,

and Tan 29 with x + y:

x = and: x + y= simplify

x = 2,546.85 and x + y = 10,824.29

So if we subtract the two from each other, we get:

y = 8,277.44, closest to choice (C).


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5)

ANSWER: (C) 14 cm

Did you get it?

Draw the original with more labels:

As you can see, I took side AC, and divided it into two equal

parts since thats what the problem said to do. I did not,

however, divide the ABC angle into two equal parts. Just

because a line divides a side of a triangle in half, doesnt

mean it divides the angle in half! It also doesnt make a right

angle with side AC. So if you made one of those two

mistakes, learn now!

Next, we need to figure out which part of the triangle to solve

for first. Well, we have the angle and its opposite side, so we

can use that to find angle BAC while using its opposite side,

using law of sines:

= Cross multiply

17 (sin A) = 12 (sin 58) Divide both sides by 17

Sin A= Simplify

Sin A = 0.598622 Take the arcsin of both sides

Angle A = 36.77o

OK, thats the first part. And since we now know two angles,

we can subtract those angles from 180 and get angle C.

Angle C = 85.23o

IV. I saw we re-draw the triangle:

Since we are trying to find BD, we must look at the botto

triangle. We have two sides (8.2, 12) and an angle (85.2

We can use the law of cosines to find that last side!

V. BD2= 12

2+ 8.5

2(2 12 8.5 cos 85.23) Do the

parenthesis fi

BD2= 12

2+ 8.5

2(16.96384)Do the right side.

BD2= 199.286Square root of both sides

BD = 14.12, which is closest to choice (C).

If you like this test, maybe you would be intereste

my full 0061 practice exams, available at

www.praxis2math.comfor $5.00. The practice exa

consists of 25 questions of various mathematical

applications, complete with a second exam of 25

questions of similar type. The idea is to take the f

exam, work out your answers, learn from your

mistakes, and improve the second time around.

Or if you would like to contact me, please do at:

[email protected]. Im pretty quick to respond

dont hesitate!
http://www.praxis2math.com/http://www.praxis2math.com/mailto:[email protected]:[email protected]:[email protected]://www.praxis2math.com/

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