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.
Math 3 Trigonometry Part 1 Unit Updated July 26, 2016
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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.
Math 3 Trigonometry Part 1 Unit Updated July 26, 2016
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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?
Math 3 Trigonometry Part 1 Unit Updated July 26, 2016
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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
Math 3 Trigonometry Part 1 Unit Updated July 26, 2016
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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