Download Course Correlation - York County School Division

Course Correlation to Virginia Standards of Learning
Name of Provider:
York County School Division
Name of Course:
Trigonometry
URL for Course Syllabus: http://yorkcountyschools.org/virtualLearning/default.aspx
Last Revision Date:
2010-11 (Courses revised quarterly as needed)
Trigonometry SOL
Standard:
How does the course content address this
standard? (Please refer to the syllabus
posted on your Web site.)
T.1
The student, given a point other than the Trigonometric Functions
origin on the terminal side of an angle,
will use the definitions of the six
trigonometric functions to find the sine,
cosine, tangent, cotangent, secant, and
cosecant of the angle in standard
position. Trigonometric functions
defined on the unit circle will be related
to trigonometric functions defined in
right triangles.
T.2
T.3
The student, given the value of one
trigonometric function, will find the
values of the other trigonometric
functions, using the definitions and
properties of the trigonometric functions.
The student will find, without the aid of a
calculator, the values of the
trigonometric functions of the special
angles and their related angles as found
in the unit circle. This will include
converting angle measures from radians
to degrees and vice versa.
 Basic Concepts and Terminology
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o Recognize and draw an angle in standard
position.
– Calculate the values angles coterminal to a
given angle in standard position.
Radians
o Recognize various systems used to measure
angles, including degrees, radians, gradians,
and mils.
– Use a conversion factor to convert from
radians to degrees and vice versa without
using a calculator.
Right Triangle Trigonometry (acute angles)
o Define the six trigonometric functions of an
angle in a right triangle.
– Calculate the values of the six trig functions
of the acute angles, given the measures of
any 2 sides of a right triangle.
Trig With Non-Acute Angles
o Define the six circular trigonometric functions
of an angle in standard position.
– Relate the coordinates of a point on the
terminal side of an angle to the position
and dimensions of an appropriate
reference triangle.
– Calculate the values of the six trig functions
of any angle (including non-acute), given
the coordinates of a point on the terminal
side of the angle.
o Calculate the other five trigonometric function
values of an angle, given one trigonometric
function value and the quadrant in which the
angle lies,.
– Relate the definitions of the six trig
functions to the position and dimensions of
an appropriate reference triangle.
Special Angles
o Develop the unit circle, using both degrees
and radians, recognizing and using the
various patterns embedded in the circle.
– Know/list (memorize) the measures of the
special (and quadrantal) angles in both
degrees and radians.
– Know/list (memorize) the trigonometric
function values of the special (and
quadrantal) angles and their related angles
in both degrees and radians.
– Solve problems using the circular function
definitions and the properties of the unit
circle.
o Recognize the connections between the
coordinates of points on a unit circle and
– coordinate geometry;
– cosine and sine values; and
– lengths of sides of special right triangles
(30°-60°-90° and 45°-45°-90°).
 Inverses of the Special Angles
o Calculate the measures of angles when their
trigonometric function values are given.
– Calculate the trigonometric function values
of any angle in either degrees or radians.
T.9
The student will identify, create, and solve
practical problems involving triangles.
Techniques will include using the
trigonometric functions, the Pythagorean
Theorem, the Law of Sines, and the Law of
Cosines.
T.4
The student will find, with the aid of a
calculator, the value of any trigonometric
function and inverse trigonometric function.
Applications
 Using Calculators in Trig
 Solve Right Triangles
o Solve a right triangle.
– Calculate all three unknown measurements
of any right triangle, given the measures of
at least one side and one other dimension
(angle or side).
– Use the Pythagorean Theorem, trig
functions, and inverse trig functions to
calculate the unknown quantities.
– Round calculated values to the appropriate
number of significant digits, based on the
precision of the given measurements.
– Sketch diagrams of problems and correctly
position values and variables (angles
opposite appropriate sides).
 Applications
o Solve practical problems (applications)
involving triangles.
– Sketch diagrams from word problems,
placing values and variables in appropriate
positions; ensure a correctly place right
angle to allow solution of the problem using
right triangle trig techniques.
– Correctly place horizon lines to establish
angles of elevation and/or depression for
vertical application problems.
– Correctly establish the right angle for
horizontal application problems.
 Law of Sines, Ambiguous Case, Law of Cosines
o Solve oblique (non-right) triangles, using
combinations of the trigonometric functions,
Pythagorean Theorem, Law of Sines, Law of
Cosines, and the Angle Sum Theorem.
– Identify the most efficient solution
technique to be used with a given problem,
based on the information (angle and side
measures) given.
– Law of Sines:
 Recogize as an extension of the
definitions of the trig functions (SOHCAH-TOA) for oblique triangles,
 Requires a pair of opposites: angle and
side
 Use when given AAS or ASA (with the
Angle Sum Theorem)
– Ambiguous Case of Law of Sines:
 Use when given SSA.
 Determine how many triangles exist for
the given starting information.
– Law of Cosines:
 Recogize as an extension of the
PythagoreanTheorem for oblique
triangles,
 Finds an opposite: angle or side
 Use when given SAS or SSS
information.
 Recognize when and how to use the
Law of Sines to determine follow-on
values without falling into the
Ambiguous Case.
– Pythagorean Theoren and Angle Sum
Theorem:
 Use to calculate required starting values
for the 2 Laws,
 Use to find the measure of the 3rd angle
of a triangle.
o Calculate the area of any triangle.
– Use the Laws of Sines and/or Cosines to
calculate needed side lengths.
o Calculate the altitude of a triangle from a
known base.
T.6 The student, given one of the six
trigonometric functions in standard
form [e.g., y = A sin (Bx + C) + D,
where A, B, C, and D are real
numbers], will
a) state the domain and the range of the
function;
b) determine the amplitude, period, phase
shift, and vertical shift; and
c) sketch the graph of the function by
using transformations for at least a oneperiod interval.
Graphing Trig Functions
 Graphing Sine and Cosine (Amplitude and
Period)
 Translating Sine and Cosine Graphs
 Equations from Graphs
 Cosecant and Secant Graphs
 Tan and Cot Graphs
o Graph sin and cos curves.
– Use the special angles in a table of values
to craft graphs of the parent sine and
cosine functions.
– Expand the table of values (rows) to create
graphs of transformed functions.
– Recognize and identify the key points on
the sine and cosine graphs; locations of :
 zeros (crossing points on the shifted xaxis)
 local maximum and minimum points,
based on amplitude from the shifted xaxis.
– From an equation in standard form, sketch
a graph using the key-points method.
o Graph sec, csc, tan, cot curves.
– Use the special angles in a table of values
to craft graphs of the parent functions.
– Expand the table of values (rows) to create
graphs of transformed functions.
– Recognize and identify the key points on
the graphs; locations of :
 zeros (crossing points on the shifted xaxis)
 local maximum and minimum points,
based on amplitude from the shifted xaxis.
 Asymptotes
 Amplitude check points (tan and cot)
– From an equation in standard form, sketch
a graph using the key-points method.
o Write the equation of the trig function, given
its graph.
o From the graph, determine critical
information, including shifted origin, phase
and vertical shifts, amplitude, and period.
T.5
T.1
The student will verify basic trigonometric
identities and make substitutions using the
basic identities.
The student, given a point other than the
origin on the terminal side of an angle, will
use the definitions of the six trigonometric
functions to find the sine, cosine, tangent,
cotangent, secant, and cosecant of the angle
in standard position. Trigonometric functions
defined on the unit circle will be related to
trigonometric functions defined in right
triangles.
Trigonometric Identities
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Fundamental Trig Identities
Proving Trig Identities
Sum and Difference Identities
Double-Angle Identities
Half-Angle Identities
o Use the fundamental identities to find the
values of the other 5 trig functions, given the
value of one function.
– Express one function in terms of another..
– Recognize the connection and equivalence
between this use of the fundamental
identities and the right triangle trig work
done in Unit 1 using the definitions of the 6
trig functions along with the Pythagorean
Theorem.
o Write equivalent expressions.
– Use the fundamental identities to rewrite an
expression in terms of only sine and
cosine.
– Apply other standard algebra tools,
including factoring.
o Simplify trig expressions.
– Use the fundamental identities to combine
terms in an expression, with the goal of
achieving a single term with a single trig
function in the numerator.
o Verify trigonmetric identities.
– Prove that a trig equation is valid (true) for
all meaningful values of  (for every value
in their domains).
– Given an trig equation, use the trig
identities so simplify each side of the
equation independently to the point where
both sides of the equation are identical.
o Calculate exact values for the sine and
cosine of the sum or difference of two angles.
– Rewrite a given angle as the sum or
difference of two special angles.
– Use the sum and difference identities,
along with the special angles, to calculate
exact values.
– Given a single trig function value of each of
two angles, use right triangle trig tools
(from Unit 1) to determine the sine and
cosine of the angles, then use the
appropriate identities to calculate the sine
and cosine of the sum or difference of the
given angles.
o Calculate exact values for the sine and
cosine of twice or half of a given angle.
– Recognize the double-angle identity as a
simplification of the sum identities, when
the two angles are the same. As such, the
sum identity can always be used in its
place.
– Recognize the half-angle identity as a
simplification of the cosine sum identity,
when the two angles are the same. As
such, the cosine sum identity can always
be used in its place when worked
“backwards”.
– Given a single trig function value of an
angle, use right triangle trig tools (from Unit
1) to determine the sine and cosine of the
angle, then use the appropriate identities to
calculate the desired sum.
G.5, G.6
Unit Objectives:
The Learner Will:
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
given the lengths of three segments, determine
whether a triangle could be formed.
arrange the angles of a triangle in order from
smallest to largest when given the lengths of
the sides.
Inverse Trig Functions and Trig
Equations
 Inverse Trig Functions
 Solving Trig Equations
o Find the measures of angles in right triangles.
– For the special angles, determine the angle
measure when given a trig function ratio
and quadrant for a point on the unit circle.
 Use coterminal angle concepts to
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arrange the sides of a triangle in order from
smallest to largest when given the measures of
the angles.
given the lengths of two sides of a triangle,
determine the range in which the length of the
third side must lie.
determine all possible values for the
angle.
– Use the inverse trig functions to find angle
measures when either:
 The sides of a right triangle are know, or
 When a trig function value is known,
along with its quadrant.
 Determine all possible values for the
angle.
– Calculate the Principal Value for an inverse
trig function.
 Sketch and identify the graphs of the
inverse of each trig function.
 Determine the domain and range
limitations of each in order for the result
to remain a function.
 Use the range restrictions to calculate
the Principal Value (also sometimes
refered to as the Calculator Value or
Function Value)
 Convert the Principal Value into an
appropriate value within a given desired
range.
o Solve trigonometric equations with:
– With either:
 Infinite solutions
 Restriced rangesolutions.
– Using all trig and algebra tools learned so
far, including:
 Factoring,
 Identities.