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TRIGONOMETRY
Trigonometry has its origins in the study of triangle
measurement. Natural generalizations of the ratios of righttriangle trigonometry give rise to both trigonometric and
circular functions. These functions, especially the sine and
cosine, are mathematical models for many periodic real world
phenomena. Students studying trigonometry should explore
data from such real world phenomena, and also should identify
and analyze the corresponding trigonometric models. The study
of inverse trigonometry functions; trigonometry equations and
identities; the Law of Sines and the Law of Cosines; vectors;
and polar coordinates should also be included in the course.
It is important to note that all of the objectives and
competencies listed in this outline have also been included in
the list of competencies for Pre-Calculus.
Please refer to the general introduction to the Indiana High
School Mathematics Competencies (pp. i - iv) for more
information on the structure of the competency lists, modes
of instruction and the use of technology.
Scientific calculators and graphing technology play an
important role in the development of students' understanding
of the concepts of trigonometry as well as providing students
with more time and power to explore realistic applications.
Trigonometry - 162
OBJECTIVE 1
The learner will demonstrate proficiency in using scientific calculators and graphing
technology.
Competency
1.1
Graph functions and relations.
1.2
Analyze functions. (Zeros, domain, range, asymptotes, etc.)
1.3
Solve equations.
1.4
Evaluate trigonometric and inverse trigonometric expressions.
1.5
Verify trigonometric identities.
1.6
Convert radians/degrees.
Trigonometry - 163
OBJECTIVE 2
The learner will apply trigonometry to problem situations
Competency
2.1
Solve real world
problems involving right
triangle trigonometry.
Example
2.1.1 You are given the job of measuring the height of a mountain. Standing on level
ground, you measure the angle of elevation to the top to be 17.3 degrees. You now
move 350 meters closer and find the angle of elevation to be 29.1 degrees. How high
is the mountain?
2.1.2 A helicopter is approaching Indianapolis International Airport at an altitude of 3 km.
The angle of depression from the helicopter to the airport is 15.25 degrees.
a)
b)
2.2
Define trigonometric
functions using right
triangles.
2.3
Apply the laws of sines
and cosines to the
solution of application
problems.
What is the helicopter's ground distance to the airport?
What is the distance the helicopter will actually travel as it descends along the
line of sight?
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Competency
2.4
Define vectors and use in
solving real world
problems.
Example
2.4.1 Bill's ship is traveling on a bearing of 112 degrees at 29 knots. The current is on a
bearing of 200 degrees at 7 knots. Find the actual bearing and speed of the ship.
2.4.2 A plane is flying on a heading of 310º at a speed of 320 miles per hour. The wind is
blowing directly from the east at a speed of 32 miles per hour.
a)
b)
c)
d)
e)
f)
g)
Make a drawing to indicate the plane's motion.
What equations model the plane's motion (without the wind)?
Make a drawing to indicate the wind's motion.
What equations model the wind's motion?
What are the resultant equations modeling the plane with the wind?
After 5 hours, where is the plane?
Carefully describe how to find the angle adjustment the pilot must make so that the
plane is not blown off course during a 5 hour flight.
h) Write the final equations which combine both plane and wind motion
contributions to the flight.
i) Test your equations by graphing and tracing.
Trigonometry - 165
OBJECTIVE 3
The learner will develop an understanding of trigonometric functions.
Competency
3.1
Solve real world
problems involving
applications of
trigonometric functions.
Example
3.1.1 A creature from Venus lands on Earth. Its body temperature varies sinusoidally with
time. Twenty-five minutes after landing it reaches a high of 130 degrees Fahrenheit.
Fifteen minutes after that it reaches its next low of 100 degrees Fahrenheit.
a)
b)
c)
d)
3.2
Develop the relationship
between degree and
radian measures.
3.3
Define trigonometric
functions using the unit
circle.
3.4
Learn exact sine, cosine,
tangent values for
multiples of
B,
Sketch the graph of this function.
Write an equation expressing temperature in terms of minutes after landing.
What was the temperature upon landing?
Find the first two times the temperature was 117 degrees.
3.2.1 Hold your arm out at eye level. Have someone measure the distance from your eye to
the end of your fingertips. Cut a strip of paper that length. Now, with your arms
outstretched, hold the paper parallel to your body, with one hand on each end of the
strip. The angle your arms form approximates a radian. Why? Approximately how
many degrees is your angle?
B B B B
, , , , 0.
2 3 4 6
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Competency
3.5
Find domain, range,
intercepts, periods,
amplitudes, and
asymptotes of the
trigonometric functions.
3.6*
Identify odd and even
functions and the
implications for their
graphs.
Example
* Optional
3.7
Graph trigonometric
functions and analyze
graphs of trigonometric
functions.
3.8
Define and evaluate
inverse trigonometric
functions.
3.9
Analyze and graph
translations of
trigonometric functions.
3.7.1 Graph y = 4 - 3 cos (2x - 40). Identify the period, amplitude, phase shift and vertical
displacement.
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Competency
3.10
Make connections among
the right triangle ratios,
trigonometric functions,
and circular functions.
Example
3.10.1 Given the right triangle below,
a) find the exact sine, cosine and tangent of angle A.
b) find the real numbers x, 0 # x < 2B , with exactly the same sine, cosine, and
tangent values.
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OBJECTIVE 4
The learner will solve trigonometric equations and verify trigonometric identities.
Competency
4.1
Solve real world problems
involving applications of
trigonometric equations.
Example
4.1.1 A baseball player hits the ball and the ball leaves the bat at an angle of x with a
velocity, v, of 100 ft/sec. The ball is caught 350 ft away from home plate. Use the
equation below to find the angle of projectile (x) of the ball.
1
( V 2 @ sin2x )
32
4.1.2 A gun with a muzzle velocity of 1200 ft/sec is pointed at a target 2000 yards away.
Excluding air resistance, what should be the minimum angle of elevation of the gun?
d=
d=
4.2
Apply the fundamental
trigonometric identities.
4.3
Use the fundamental
identities to verify simple
identities.
1
( V 2 @ sin2x )
32
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Competency
4.4
Use graphing technology
for solving trigonometric
equations.
Example
4.4.1 Use a graphing calculator to solve for real numbers, x, 0 # x <2B. Discuss the results
of your findings.
a)
b)
c)
d)
cos x = -1
cos 2x = -1
cos 3x = -1
6 sin 2x = 5 (y = 6 sin 2x and y = 5)
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OBJECTIVE 5
The learner will understand the connections between trigonometric functions and polar
coordinates and complex numbers.
Competency
5.1
Define polar coordinates and relate polar to Cartesian coordinates.
5.2
Graph equations in the polar coordinate plane.
5.3
Define complex numbers and convert to trigonometric form.
5.4*
State, prove, and use DeMoivre's Theorem.
* Optional
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