TRIGONOMETRY2011
Grade 11-12
ANGULAR RELATIONSHIPS
CONTENT STANDARD: Students will understand the concepts related to standard position and radian measure and be able to apply these concepts to solve mathematical problems.Big Ideas and Essential Questions
The student understands that…
In order to draw an angle in standard position, one must know its direction, initial and terminal side locations.
What does it mean to draw an angle in “standard position?”
How does one identify an angle’s direction in standard position?
The radius of a circle and π has a direct relationship to radian measure.
How does the radius of a circle and “π” relate to the concept of radian measure?
How are angles measured and converted inboth radians or degrees?
Radian measure and circles have a direct connection to the study of circular motion.
How can someone minimize their chances of getting motion sickness on a merry-go-round and how does one maximize there chances of “getting the most out of their ride?”
How can the length of an arc, an angle’s displacement, and the area of a sector be computed?
Grade Level Expectations
The student is able to…
a.Draw angles in standard position by accurately drawing an initial and terminal side.
b.Show direction and rotations of an angle in the both positive and negative direction.
c.Derive the concept of radian measure from the circumference formula (C=2πr).
d.Approximate and draw any angle given in either degree or radian measure.
e.Convert angle measurements from one format to the other (degrees – radians).
f.Find angles that are co-terminal with each other in both formats.
g.Use the arc length formula (s = rθ) and area of a sector formula (A= ½ r2θ) in real world problem settings.
h.Calculate the angular and linear velocities of various real-world scenarios.
Key Vocabulary: initial side, terminal side, standard position, central angle, displacement, radian, and the Greek alphabet.
Instructional Strategies: See Appendix for suggested instructional strategies.
TRIGONOMETRIC FUNCTIONS (Circular Functions)
CONTENT STANDARD: Circular functions and their use are the basis of the field of Trigonometry.Big Ideas and Essential Questions
The student understands that…
The Pythagorean Theorem is a useful tool in finding the exact values of the Unit Circle.
How is the Unit Circle constructed and its special angles identified using the Pythagorean Theorem and the special 45-45-90 or 30-60-90 triangle?
There are fixed relationships between the angles and lengths of the sides of the six trigonometric functions are essential to the study of trigonometry.
How is sine, cosine, tangent (and their co-functions) derived?
How can any of the trigonometric functions be evaluated for any angle given a point on its terminal side?
How can an exact angle measure be found, given a trigonometric value?
In order to work with non-unit circle angle measures, the use of a scientific calculator is essential.
How can a scientificcalculator be used to approximate any of the six trigonometric functions when given an angle measure?
How can the scientific calculator be used to approximate any angle measure when given a trigonometric function’s value?
Grade Level Expectations
The student is able to…
a.Compute the sides of a 45-45-90 and 30-60-90 triangle, given that the hypotenuse is 1.
b.Identify the exact values of all six trigonometric functions at the known locations on the unit circle.
c.Draw a unit circle and identify the 16 known special locations in both radians and degrees.
d.Identify the exact values of all six trigonometric functions at any known location within the Cartesian coordinate system.
e. Identify the exact value of any known angle measure, given its exact trigonometric value.
f.Use the trigonometric definitions to find the exact values of any function, given the value of one of them.
g.Use a scientific calculator (in both degree and radian mode) to find the approximation of any trigonometric function, given its angle measure.
h.Use a scientific calculator to approximate the possible angle measure(s) when given any trigonometric value.
Key Vocabulary: cosine, sine, tangent, cotangent, secant, cosecant, reference triangle.
Instructional Strategies: See Appendix for suggested instructional strategies.
TRIGONOMETRIC FUNCTIONS (Fundamental Identities)
CONTENT STANDARD: The Fundamental Identities have useful meaning in the study of Trigonometry.Big Ideas and Essential Questions
The student understands that…
Trigonometric functions can be evaluated with proper use of the fundamental trigonometric identities.
How can the fundamental trigonometric identities be used to evaluate or verify other trigonometric identities?
Trigonometric expressions can be simplified by proper use of the fundamental identities.
How can trigonometric expressions be simplified using the fundamental identities?
The fundamental trigonometric identities can be used to develop additional trigonometric identities.
How can trigonometric identities be used to develop and prove other identities?
Grade Level Expectations
The student is able to…
a.Verify trigonometric identities by using definitions (x, y, and r).
b.Verify trigonometric identities by using exact and non-exact trigonometric values.
c.Use known identities (and algebra rules) to help simplify trigonometric expressions.
d.Use the fundamental identities (and algebra rules) to prove other trigonometric identities.
Key Vocabulary: fundamental identities, odd-even identities, Pythagorean identities, ratio identities, and reciprocal identities.
Instructional Strategies: See Appendix for suggested instructional strategies.
GRAPHS OF TRIGONOMETRIC FUNCTIONS
CONTENT STANDARD: The graphs of the six trigonometric functions allow for deeper a deeper understanding of their properties.Big Ideas and Essential Questions
The student understands that…
The trigonometric functions are periodic.
How are the trigonometric functions periodic functions?
Graphs of the trigonometric functions may be transformed and translated.
How can a basic trigonometric function’s graph be relocated anywhere on the Cartesian Coordinate System?
Simple harmonic motion problems in the real word are modeled by using sinusoidal functions.
How can you determine your height off the ground at any time while riding a Ferris wheel?
How can the rotation of the planets around the sun be modeled by a sinusoidal function?
Grade Level Expectations
The student is able to…
a.Graph and label any trigonometric function over a fixed interval.
b.Determine the domain and range of a given function’s equation or graph.
c.Identify the amplitude, period length, phase shift, horizontal shift, asymptote locations and zeros (when applicable) of any trigonometric function.
d. Utilize the characteristics of the sine and cosine functions to develop real-world models, and graph, simple harmonic motion problems.
Key Vocabulary: Amplitude, period length, phase shift, vertical shift, asymptote lines, zeros of a function, simple harmonic motion, and sinusoidal functions.
Instructional Strategies: See Appendix for suggested instructional strategies.
APPLICATIONS OF TRIGONOMETRY
CONTENT STANDARD: Trigonometric relationships can be used to solve all types of “right” and “non-right” triangular scenarios in the real world.Big Ideas and Essential Questions
The student understands that…
Indirect measurements of lengths and angles of a triangle, in a real-world setting, can be arrived at with proper use of trigonometric relationships.
What heading would a US Air Force F-16 fighter jet have to take to quickly intercept an unauthorized aircraft in their vicinity?
Without climbing a tree near your house, how does a tree cutter determine whether he can safely cut it down without damaging any property?
Grade Level Expectations
The student is able to…
a.Define the trigonometric functions in terms of the sides of a right triangle.
b.Find the missing angle(s) or side(s) in a right triangle using the trigonometric functions.
c.Find the missing angle(s) or side(s) in a non-right triangle by using the Law of Sines and the Law of Cosines.
d.Find the area of a non-right triangle by using the formula: K = or
(Heron’s Formula) .
Key Vocabulary: opposite side, adjacent side, hypotenuse, ambiguous case, SAS, ASA, SSS Postulates.
Instructional Strategies: See Appendix for suggested instructional strategies.
TRIGONOMETRIC IDENTITIES
CONTENT STANDARD: Additional identities can be used to solve equations and inequalities, and further develop an understanding of more complex trigonometric concepts.Big Ideas and Essential Questions
The student understands that…
Solving trigonometric equations and inequalities require manipulation of trigonometric identities.
How can one solve equations or inequalities that contain trigonometric functions?
Sum and difference formulas for sine, cosine, and tangent, as well as multiple-angles and product to sum formulas help simplify expressions and solveproblems.
How can you find the exact value of a trigonometric function on a unit circle at a location that is not one of the original 16 known locations?
How can you simplify expressions and solve problems that were before, “unsolvable,” by using the multiple-angle and product to sum formulas?
Grade Level Expectations
The student is able to…
a.Solve trigonometric equations and inequalities.
b.Find additional exact values on the unit circle by using the sum, difference, double- and half- angle formulas.
c.Apply the product-to-sum formulas to simplify expressions and solve equations.
Key Vocabulary: Sum & difference formulas, double- half- formulas, product-to-sum formulas.
Instructional Strategies: See Appendix for suggested instructional strategies.
POLAR COORDINATES
CONTENT STANDARD: The polar coordinate system is a graphing system which requires the knowledge and application of trigonometry.Big Ideas and Essential Questions
The student understands that…
The polar coordinate system is a two-dimensional system in which each point is determined by a distance from a fixed point (pole) and an angle from a fixed direction.
How can you construct a dart board and locate the position of the thrown darts by using polar graphing techniques?
How can you create a graph that looks like a “flower?”
The Cartesian Coordinate System and the Polar coordinate system have related characteristics.
How can you convert a polar coordinate to a more familiar, Cartesian coordinate?
Grade Level Expectations
The student is able to…
a.Plot polar coordinates and identify them using the correct notation of (r, θ).
b.Change Cartesian coordinates to polar coordinates or polar to Cartesian form.
c.Graph basic polar equations.
Key Vocabulary: pole, directed distance, polar coordinate.
Instructional Strategies: See Appendix for suggested instructional strategies.