Download Class Exercise06 Basic Trig Khan Academy With Figures

Name: ___________________________
Introductory Exercise 06 – Basic Trigonometry and Calculator Functions
Allotted Time: 2.5 hours (come to “Help Session” if one is scheduled)
The introduction to Trigonometry is easiest when extended from Geometry, in
particular from the definition of a “right triangle”. We will not delve into basic
Geometry. If you are not familiar with the definition of “right triangle” (a triangle with
a 90-degree angle) and Pythagorean theorem, please proceed to learn from the
following video, and all subsequent videos if needed.
https://www.khanacademy.org/math/cc-eighth-grade-math/cc-8th-geometry/cc8th-pythagorean-theorem/v/pythagorean-theorem
Trigonometry at the most elementary level is easily understandable by the relationship
between the 3 sides and the 3 angles of right triangles. You will become reasonably
familiar with the three basic trigonometric functions: sine, cosine and tangent
functions. You will not be concerned with secant, cosecant and cotangent functions.
If you are not familiar with the 3 basic trigonometric functions, please proceed to Khan
Academy to begin learning:
https://www.khanacademy.org/math/trigonometry
Go to “Basic Trigonometry” and start with the first video and proceed to learn from
all subsequent videos until you become familiar with sine, cosine and tangent
functions. You should at least try to understand the few figures, and pick up on the
moniker “SOH CAH TOA”. A quick summary is provided below.
Summary:
Refer to Figure 1 above that displays a right triangle. The angle between side a and
side b is the “right angle”, which has 90 degrees – meaning that side a and side b are
1
Name: ___________________________
“perpendicular”. The angle “theta”, denoted as , is used in this case to label the
angle between sides h and a. The side a touching the angle  is referred to as the
“adjacent side” (to ). The side opposing the angle is referred to as the “opposite
side” (to ), which is b in this case. Hypotenuse h is the side opposing to the right
angle and is always the longest side of the triangle. Please note again that what is
“opposite” and “adjacent” depends on the angle you are looking at, and they will be
different for different angles. Now let’s relate this to basic Geometry, in which the
relationship between the numerical values of the three sides is given by the
Pythagorean Theorem:
a2 + b2 = h2
1)
Trigonometry uses the language of “angle” for numerical tabulation:
- Sine of angle “theta” is the opposite side over the hypotenuse (b over h)
Sin( ) = b / h
- Cosine of angle “theta” is the adjacent side over the hypotenuse (a over h)
Cos() = a / h
- Tangent of angle “theta” is the opposite side over the adjacent side (b over a)
Tan() = b / a
Part A) For the following practices, refer to Figure 1 on previous page:
Practice 1) Now look at the angle  (“gamma”) in Figure 1. For this angle, what is
the adjacent side to ? It is the same as the adjacent side to?
Practice 2) Similarly, what is the opposite side to ? It is the same as the opposite
side to?
2
Name: ___________________________
Practice 3) What about the hypotenuse? Is the hypotenuse different or same for
these two different angles?
Practice 4) What is “sine of gamma”, sin(), in terms of the sides? That is, write out a
trigonometric relationship, similar to ones in videos or Summary, relating the angle and
two sides (in a, b or h). Simply follow the definition base on what is “opposite” and
“adjacent”.
Practice 5) Similarly, what is “cosine of gamma”, cos()?
Practice 6) What is “tangent of gamma”, tan()?
Practice 7) If you are told that a= 6 and b=7, then what/which trigonometric
function(s) can you immediately work out (meaning, calculate a number and know its
numerical value) just by following the definition you have learned?
Part B)
One of the great confusions about trigonometric functions is that users do not
understand that these functions have been tabulated and worked out for all values of
angles – along with inverse functions (to be learned later when time permits) they are
to be used alongside known information about the sides of the triangle to solve basic
3
Name: ___________________________
geometric or algebraic problems.
The two primary types of basic math problems involving a trigonometric function
involve
2 given information to find another:
1) Using a given angle and a given side to find the value of another side
2) Using two sides to find an angle – this will involve the inverse trigonometric
function.
In both cases, you will need to use a calculator (or such things as Google Calculator
online) – in the old time, mathematicians use a tabulated table. The first thing is you
will need to know is that, Radian and Degree are two different scales to express the
numerical values of angles – Degree goes from 0 to 360, and Radian goes from 0 to
2*pi (0 to 6.2832…). They are, again, like different units expressing the same
physical quantities. Any calculator can compute a trigonometric function in either
Degree or Radian mode. You need to know which one your calculator is using.
Practice 8) Check the “mode” of your calculator by punching in cos(90). If the
calculator displays 0.000000 then the mode is in “degree”; if the calculator displays
0.44807 then the mode is in “radian”. Ensure to set the mode to “degree”. We will
use “degree” to practice, but “radian” is also important if not far more so. (*You may
need to find out how to set radian/degree mode in your calculator – simple calculator
typically has a “mode” or simply “rad/deg” button. For TI-85 or similar please use
the following video: https://www.youtube.com/watch?v=y-IbbJYG4CA)
In “degree” mode, find sin(90º) and tan(45º). The answer is a very simple number.
Practice 9) Practice: find cos(30º) and cos(390º), sin(265º), etc.
Practice 10) Google Practice: go to Google (www.google.com), and in the search or
URL simply type in the expression sin(45). A google calculator will be displayed along
4
Name: ___________________________
with a number.
Is it in “radian” or “degree”? Now do a few more like tan(45),
cos(30), and so on. What “mode” does Google calculator adopt as the default (how
can you tell)?
Practice 11) Instead of typing in sin(45), type in sin(45 degree) or sin(45) degree in
the search box or URL or anything reasonable telling Google that you want it in
“degree” mode. What do you get? Now try sin(45 radian), and what do you get?
Practice 12) Using Google, tabulate cos(90) using degrees and then sin(3.14159) in
radian. BTW, what famous number is 3.14 that you may recall? Try sin(pi radian),
cos(pi radian) and tan(pi radian).
Now as you are familiar with the definition of sine, cosine and tangent, and know how
to use a calculator to find the value of these trigonometric functions for a given angle
(in radian or degree), let’s proceed with the first type of problem – using one angle
and one side to determine another side. Use the following Khan Academy lesson (in
“Basic Trigonometry”) as an example and a warm-up.
https://www.khanacademy.org/math/trigonometry/basic-trigonometry/basic_trig_ratios/v/exampletrig-to-solve-the-sides-and-angles-of-a-right-triangle
(In this example, again be reminded that for a given angle, the tangent of that angle is
evaluated using a calculator, as Salman Khan shows in the video – you are not to
evaluate the tangent of a known angle using the sides as it may appear deceptively so
from the definition of the trigonometric function! In this case, even if you want to use
the sides, you cannot, because one side is not given – you are instead asked to solve
5
Name: ___________________________
for the unknown side, and that is why you need to know the numerical value of the
trigonometric functions first. This is often the major hurdle that confuses the
beginning learners.)
Practice 13) Please refer to Figure 1 again. If  is 30 degrees, and b = 10, then what
is a? (First think about which of the three trigonometric functions involve the three
quantities in question – , a and b, write out the expression, use calculator to evaluate
the trigonometric function for the given angle, and finally solve for a algebraically.)
Derek Owen has a set of great examples. Enjoy other videos at your own choosing.
There is no shortage of good lessons you can find online.
https://www.youtube.com/watch?v=lbTs7MCfPbI&list=PL92C1891EA55C6A0A&index=39
Practice 14) Using Figure 1 and the same set up as the previous practice,  = 30
degrees, and b = 10, but instead what is h? (Again it is the same steps – determine
which trigonometric function to use, write down the expression, use calculator to
evaluate the trigonometric function of the given angle numerically, and finally solve for
the side algebraically).
6
Name: ___________________________
7
Name: ___________________________
Part C) Preview to the next learning exercise
The second type of problem involves in determining an angle if you have information
about length or distance (but not angle). For example, in Figure 1, if you know a =
10 and h = 15, then what is angle ?
 You know the proper trigonometric function to use is cosine because it relates
“opposite” and “hypotenuse”
cos() = a/h = 10/15 = .6667
(round off to 4 decimal places)
You now have the expression cos()=.6667. The proper interpretation is “what is the
value of when taken the cosine of the value, would give .6667?” This is where you
will need to use “inverse trigonometric functions.” However, rather than to go
through some formal introduction such as a unit circle (many learning videos do just
that), you simply need to know, at this level, how to use the calculator correctly.
https://www.youtube.com/watch?v=S4XXHhhDhgM
You simply use the “Inverse Trigonometric function” on your calculator. The way you
will express this properly is to translate the expression into the following form:
cos() = .6667   = cos
-1
(.6667)
( or arcos(.6667) )
1) Think of cos -1 simply as a notation, and a more proper notation is “arccos”
(arc-cosine), which stands for the inverse function. However, most calculators
and labels will simply use the “cos -1” notation, but it is absolutely not “1 over
cosine” or “reciprocal of cosine”. So repeat, it is just a notation for the
inverse function, or arc-function, and not the “reciprocal” like a fraction.
2) The answer you get will also depend on “Radian” or “Degree” mode. Both are
right, for as long as you know which mode is the final value in.
In the following Khan Academy video series, under “Inverse trig functions” grouping,
click on the video “Example: Calculator to evaluate inverse trig function”
https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func
https://www.khanacademy.org/math/trigonometry/unit-circle-trigfunc/inverse_trig_functions/v/example-calculator-to-evaluate-inverse-trig-function
8
Name: ___________________________
Practice 15) Use your calculator in “Degree” mode, evaluate cos -1 (.6667) which then
will give the answer to the angle value in the expression cos() = .6667. Also
evaluate the same expression in “Radian” mode.
Practice 16) Google Calculator provides the more clear and confident result once
again. Simply type in an expression in the search box or in the URL field “calculate
arccos(.6667) in degree”, or something similar, to get the answer. Note that you do
have to be a bit creative and clear with your expression this time simply because it can
otherwise be confusing even for the mighty Google due to so many possible different
interpretations if the expression is ambiguous.
Finally, in Google Calculator, find the answer but in “radian”. Compare both answers
to Practice 15).
In the following Khan Academy video series
https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func
under “Inverse trig functions” grouping, click on the video “Example: Calculator to
evaluate inverse trig function”:
https://www.khanacademy.org/math/trigonometry/unit-circle-trigfunc/inverse_trig_functions/v/example-calculator-to-evaluate-inverse-trig-function
Summary: In sin() = v, if v is given then to find  you simply inverse the problem to
the following form: = sin-1(v) or =arcsin(v). This procedure is identical for cosine
or tangent function,
Practice 17) You are given sin() = 0.5, write out the equivalent inversion expression
using “sin-1” or “arcsin” notation to express the angle . Find the answer in both
radian and degree.
In the same Khan academy video series, click on “Angle of sun with the ground based
on shadow” for a very common example, and also one that is a good as an astronomy
9
Name: ___________________________
problem.
Practice 18) In Figure 1, given that a=5 and b=10, then what is value of ? Make sure
you identify the correct “adjacent” and “opposite” side, and the correct trigonometric
function to apply.
10