Introduction to trigonometryparadigmmath.weebly.com/uploads/8/1/7/9/81799270/...Math 3 Trigonometry...

Math 3 Trigonometry Part 1 Unit Updated July 26, 2016

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Introduction to trigonometry

The word trigonometry comes from two Greek words. The first part of the word is from Greek trigon

"triangle". The second part of trigonometry is from Greek metron "a measure." Trigonometry is literally

the measuring of angles and sides of triangles.

For our purposes, we're going to keep things pretty simple and basic, but there are so many uses for

trigonometry. Fields that use trigonometry include astronomy, navigation (on the oceans, in aircraft,

and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability

theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number

theory, cryptology, seismology, meteorology, oceanography, many physical sciences, land surveying and

geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil

engineering, computer graphics, cartography, crystallography and game development.

For a simple example, let's look at sound waves. A basic sound wave looks like this:

The best way to represent a sound wave is with trigonometry. This is a sine wave.

Let me explain where the sine wave comes from. First we start with a circle drawn on a coordinate

plane. The radius is 1. This is called a unit circle. Next we plot a point somewhere on the circle.


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We make a right triangle with one vertex on the origin. The length of the base

leg of the right triangle is the x value of the (x,y) point. The length of the other

leg of the triangle is the y value of the (x,y) point. The hypotenuse is the radius

of the circle, which is 1. The sine value is the relationship between the sides of

this triangle. Starting from the vertex at the origin, the sine value is a ratio of

the side opposite from this angle over the hypotenuse. Sine =

. In

this case, since the side opposite is y and the hypotenuse is 1 the fraction is

really easy. Sine =

=

= y. As the point moves around the circle,

the y values increase and decrease. When you unroll the circle, the y values make a wave pattern.

We can manipulate the wave length with a little math magic.

Sound waves, light waves, sonar waves, micro waves, ocean waves, etc. all follow this pattern. It's

interesting to know that the best description for this movement comes from measuring a triangle inside

a circle. Math is the language that best describes nature and the world around us.


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The ratios of sides of a right triangle

Trigonometry is all about the relationships of sides of

right triangles. In order to organize these relationships,

each side is named in relation to an angle. Starting with

angle x, there is a side that is adjacent to that angle and

there is a side that is opposite of that angle. There is

also a hypotenuse. These names are often abbreviated

by just the first letter A, O, and H.

With this angle and these three sides there are six possible relationships. Three of them are the most

commonly used. They are called sine, cosine and tangent. These are often abbreviated as sin, cos and

tan. (Just for the record, we're only referring to acute angles right now [angles less than 90 we'll deal

with obtuse angles later).

Sine =

Cosine =

Tangent =

People often use the acronym SOHCAHTOA to help remember which is which. In the triangle below:

Sine A =

=

Cosine A =

=

Tangent A =

=

Notice that we have to refer to the angle, in this case angle A. We need to know which angle we're

starting from in order to know which side is opposite and which is adjacent.


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If we are using angle B as the reference point, then we get different answers.

Sine B =

=

Cosine B =

=

Tangent B =

=

Refer to the 3-4-5 triangle at the right for the next 3 questions.

1. What is sine X?

2. What is cosine X?

3. What is tangent X?

4. In triangle CAT, what is sine T?

5. In triangle CAT, what is cosine T?

6. In triangle CAT, what is tangent T?

7. In triangle CAT, what is sine C?

8. In triangle CAT, what is cosine C?

9. In triangle CAT, what is tangent C?


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10. In the following figure, angle B is a right angle, and the measure of angle C is What is the

value of cos (Hint: In trig, they often label an angle with this symbol This is the Greek

letter theta. Treat it as any other variable.)

11. In the right triangle below, sin = ?

12. Given the following right triangle, LMN, what is the value of cos N?


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13. For angle D in DEF below, which of the following trigonometric expressions has value

?

A. sin D

B. tan D

C. cos D

D. sec D

E. csc D

14. What is the value of sin C in right triangle ABC below?

A.

B.

C.

D.

E.

15. To determine the height h of a tree, Roger stands b feet from the base of the tree and

measures the angle of elevation to be as shown in the following figure. Which of the

following relates h and b?

A. sin =

B. tan =

C. cos =

D. sec =

E. csc =


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16. Which of the following trigonometric equations is valid for the side measurement x inches,

diagonal measurement y inches, and angle measurement w in the rectangle shown below?

A. cos w =

B. cot w =

C. sec w =

D. sin w =

E. tan w =

17. In the figure given at right, which of the following

trigonometric equations is valid (Hint: This isn't a

triangle... yet. Finish it and label it.)

A. tan =

B. cot =

C. sec =

D. sin =

E. cos =

Sometimes, we may need to use the Pythagorean theorem to find the length of the missing side or

hypotenuse in order to answer a trigonometric question. For example, in the figure below if tan =

what is sin To find sin I need

but the length of the hypotenuse isn't given. Since this is a

right triangle, I can use the Pythagorean theorem to find the hypotenuse.

25 + 144 =

=

13 = c so the hypotenuse is 13

sin

=


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18. In the following figure, sin a =

What is tan a?

19. In the figure above, sin a =

What is cos b?

20. In DEF below, DE = 1 and DF = . What is the value of tan x?

A.

B. 1

C.

D.

E. 2

21. In the following figure, tan a =

What is sin a?


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22. A group of die-hard baseball fans has purchased a house that gives them a direct view of home

plate, although their view of the rest of the field is largely impeded by the outfield wall. The

house is 30 meters tall, and their angle of vision form the top of the building to home plate has

a tangent of

What is the horizontal distance, in meters, from home plate to the closest wall

of the fans' house? (Hint: the tangent of

is a ratio. In this case it is a reduced fraction. The

adjacent side is actually 30m not 6m. Set up a ratio of equivalent fractions and solve for the

missing value.)

Ratios using an angle

So far we've worked with situations where we know at least 2

sides of a triangle. Often, we don't know 2 sides. We know

only one side and one angle. To take it to the next level, we

need to review some basics. Recall that similar triangles are

proportional. In these similar triangles at right the sides are

proportional,

=

We also recall that all right triangles have one 90 angle and if we know that one additional angle is

congruent with the angle of another triangle, those two triangles must be similar. In the figure below

angles B and E are right angles and angles C and F are congruent. That means angles A and D also have

to be congruent and these triangles are similar.


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Now referring back to the unit circle that we talked about at the beginning of this unit, we have a circle

with a radius of 1 that is centered on the origin of a standard (x,y) coordinate plane. Recall that is was

easy to find the sine of this circle because the opposite side is simply the y value and the hypotenuse is 1

Sine =

=

= .

Since it is easier to find the ratios of this circle, smart people have already figured out the sin, cos, and

tan for every angle around the circle. These numbers used to be written in books, almanacs, and tables

for people to refer to. Nowadays, they are available on calculators and computers as well. And since we

know that similar triangles are proportionate, the sine value for ANY right triangle will be the same as it

is on this unit circle. If the angle of a right triangle is 30 regardless of how big or small the triangle is,

the sine will always be

. It doesn't matter if the ratio was sine =

=

or

or

or

or

.

All these are equivalent. You can write it as

, but your calculator will display this as 0.5. Calculators will

always give the answer in a decimal form rather than the fraction, but it is still the same value. It's just

the fraction answer translated into a decimal.

Having these sin, cos, and tan values in a table, or book, or calculator, or computer, changes everything.

Now we can figure out distances when we only know one side and one angle. This is where trig

becomes really useful. This is how trig is used to calculate the distance to the stars and the

circumference of the earth and all kinds of things in astronomy, navigation, music theory, acoustics,

optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging,

pharmacy, chemistry, number theory, cryptology, seismology, meteorology, oceanography, physical

sciences, land surveying, architecture, economics, electrical engineering, mechanical engineering, civil

engineering, computer graphics, cartography, crystallography, game development, etc. But for our

purposes, we're still going to keep things simple.


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Radians

We need to introduce radians before moving on from the unit

circle. We have experience measuring with degrees. The sum of

the angles of a triangle is 180 a right angle is 90 there are 360

in a circle and so on. There is another way to measure angles in a

circle called radians. Radians are associated with the formula for

the circumference of a circle. Circumference = 2 . In the unit

circle, the radius is 1 so the formula would be circumference =

2 or simply 2 To go around a circle takes 2 radians. To

go half way around the circle is radians. To go a quarter around

the circle is

radians and so on. To convert from degrees into

radians the formula is

= radians. When you're using your

calculator to find the sine, cosine, and tangent, you need to know

if you're using degrees or radians. 30 is very different from 30

radians. Usually, the button is labeled "DRG" where you choose

the mode to be measured in degrees, radians or gradians on a

calculator. On most calculators there are small letters near the top

that indicate which mode the calculator is in. Look for the letters

DEG to indicate that it is measuring in degrees, and RAD to

indicate that the calculator is in radian mode. If you are in radian

mode and try to find the sine using degrees you will get very

wrong answers. Similarly if you are in degree mode and you try to

find the sine using radians, you will get very wrong answers.

Notice the sine wave above uses radian measurements. Try finding the sine of

using a calculator.

Make sure that the calculator is in radian mode. Change the calculator to degree mode and find the sine

of 90 These answers should be the same (both answers are 1). Now try finding the sine of using a

calculator, which mode should the calculator be in? Find sine of 180 , which mode should the calculator

be in? Are the answers the same?


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Finding missing numbers when one side and one angle are known.

In the figure below, one side is given and the measure of one angle is given. I need to solve for the

unknown value y. I notice that y is the side opposite of the given angle and I notice that 15 is the

hypotenuse. The relationship between the opposite side and the hypotenuse is sine.

Sine =

.

Sine =

=

. Plug in the values for the triangle.

sin 57 =

. The angle is 57 so my equation is complete.

15sin 57 = y. Multiply both sides by 15 to solve for y.

In many cases, we stop here and simply say y = 15sin 57 If we want to continue then we use a

calculator. Make sure it is in degree mode. Enter 57 then press the SIN button, you should get

0.838670568. That is the sine value for any triangle with an angle of 57 Multiply that by 15 to get

approximately 12.58. The length of side y is 12.58.

23. In the triangle below, what is the value of x?

A. x = 51TAN36

B. x = 51COS36

C. x = 51SIN36

D. x =

E. x =

24. Using a calculator, what is the approximate value of x

rounded to the nearest whole number?


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In the figure below, one side is given and the measure of one angle is given. I need to solve for the

unknown value x. I notice that x is the hypotenuse of this triangle. I also notice that 17 is the length of

the side adjacent to the angle. The relationship between the adjacent side and the hypotenuse is

cosine.

cos =

cos50 =

Plug in values from the triangle.

x cos50 = 17 Multiply both sides by x

x =

Divide both sides by cos50

In many cases we leave the answer like this. If we want to continue, we use a calculator. Make sure the

calculator is in degree mode. Enter 17, then press the button, then enter 50 and press the COS

button, then press the = button. Practice until you get the answer 26.44730506. This is the length of

side x.

25. In the triangle below, the angle is 40 , what is the value of x?

A. x =

B. x = 25cos40

C. x = 25sin40

D. x =

E. x =

26. Using a calculator, what is the value of x rounded to the nearest tenth?


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In the figure below, one angle is 30 and the side opposite is 5 inches. I need to find y which is the side

adjacent. The relationship between the opposite and adjacent sides is tangent.

tan =

tan 30 =

Plug in the values for the triangle.

Ytan30 = 5 Multiply both sides by Y

Y =

Divide both sides by tan30

Often we leave the answer like this. If we want to continue, we use a calculator. Enter 5, then press the

button, enter 30 then press the TAN button, then press the = button. Practice until you get the

answer 8.660254038.

27. In the triangle below, what is the value of h?

A. h =

B. h = 100tan18

C. h =

D. h = 100sin18

E. h =

28. What is the value of h rounded to the

nearest hundredth?


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29. A rock band, The Young Sohcahtoans, is trying to design a T-shirt logo. The measurements they

have chosen are represented on the figure below. The angle to the right of the logo "TYS" has

a degree measure of 35 and the side of the figure has a measure of 10 inches. Which of the

following expressions gives the measure, in inches of the diagonal top side of the figure?

A. 10 tan 35

B. 10 cos 35

C. 10 sin 35

D.

E.

30. A moving company needs a new ramp to load boxes into a truck. The loading area of the truck

is 3 feet above the ground, and they want to ramp to have a 10 angle, as shown below. At

what distance from the truck will the ramp meet the ground? Use a calculator and round to

the nearest foot.


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31. A nylon cord is stretched from the top of a playground pole to the ground. The cord is 25 feet

long and makes a 19 angle with the ground. Which of the following expressions gives the

horizontal distance, in feet, between the pole and the point where the cord touches the

ground? (Hint: if the question doesn't give you a diagram, draw one)

A.

B.

C. 25 tan 19

D. 25 sin 19

E. 25 cos 19

32. The youth center has installed a swimming pool on level ground. The pool is a right circular

cylinder with a diameter of 24 feet and a height of 6 feet. A diagram of the pool and its entry

ladder is shown below. The directions for assembling the pool state that the ladder should be

placed at an angle of 75 relative to level ground. Which of the following expressions involving

tangent gives the distance, in feet, that the bottom of the ladder should be placed away from

the bottom edge of the pool in order to comply with the directions?

A.

B.

C.

D. 6 tan 75

E. tan (6*75


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Answers

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13. C

14. C

15. B

16. D

17. A

18.

19.

20. B

21.

22. 35 meters

23. A

24. 37

25. D

26. 32.6

27. B

28. 32.49

29. D

30. 17

31. E

32. A

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