Trigonometry Item Specifications1
Reporting Category / Body of Knowledge / Discrete MathematicsStandard / Vectors
Benchmark Number / MA.912.D.9.1
Benchmark / Demonstrate an understanding of the geometric interpretation of vectors and vector operations including addition, scalar multiplication, dot product and cross product in the plane and in three-dimensional space.
Also Assesses or Assessed by / MA. 912.D.9.3
Item Types / MC
FR
Benchmark Clarification / Students will solve problems with the geometric representation of vectors including: adding vectors, scalar multiplication, unit vectors, and dot and cross products.
Content Limits / Scalars will be rational numbers only.
Stimulus Attributes / Items may be set in either real world or mathematical context. Vectors may need to be moved.
Response Attributes / Responses may be pictures or values.
Sample Item / Find the magnitude (3i+4j), where i is a unit vector along the x axis and j is a unit vector along the y axis.
Answer: 5
Reporting Category / Body of Knowledge / Discrete Mathematics
Standard / Vectors
Benchmark Number / MA.912.D.9.2
Benchmark / Demonstrate an understanding of the algebraic interpretation of vectors and vector operations including addition, scalar multiplication, dot product and cross product in the plane and in three-dimensional space.
Also Assesses or Assessed by / MA. 912.D.9.3
Item Types / MC
FR
Benchmark Clarification / Students will solve problems with the algebraic representation of vectors including: adding vectors, scalar multiplication, unit vectors, and dot and cross products.
Content Limits / Scalars will be rational numbers only.
Stimulus Attributes / Items may be set in either real world or mathematical context.
Response Attributes / Responses may not include pictures.
Sample Item / If u and v are vectors and their components are given: u = ( 3, 1) and v = (-3, 2), place the tails of the vectors together; and find the measure of the angle between vectors u and v. Round your answer to the nearest degree.
Answer: 128
Reporting Category / Body of Knowledge / Discrete Mathematics
Standard / Vectors
Benchmark Number / MA.912.D.9.3
Benchmark / Use vectors to model and solve application problems.
Also Assesses or Assessed by / MA.912.D.9.1 MA.912.D.9.2
Item Types / MC
FR
Benchmark Clarification / Students will solve real world problems involving vectors.
Content Limits / Scalars will be rational numbers only.
Stimulus Attributes / Items will be set in a real world context.
Response Attributes / Responses may include pictures or values.
Sample Item / The vector d represents the displacement of a wagon that is pulled with the force F. The work done, W (scalar), in moving the wagon in the direction of d is defined to be the component of F in the direction of d times the distance the wagon moves. Find the work done if F = (10,3) and d = (25,0).
Answer: 250
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Functions
Benchmark Number / MA.912.T.1.1
Benchmark / Convert between degree and radian measures.
Also Assesses or Assessed by / MA.912.T.1.2 MA.912.T.4.1 MA.912.T.4.2 MA.912.T.4.3
Item Types / MC GR FR
Benchmark Clarification / Missing
Content Limits / *Degrees must be stated as an integer
*Radians must be written in terms of π.
Stimulus Attributes / Items may be set in real world or mathematical context
Response Attributes / *MC responses may be expressed as degrees or radians
*GR & FR must be stated as degrees
*Responses must be in simplified form
Sample Item / In terms of π, what is the radian measure of 135 dgrees?
A. 9π
B. π/4
C. 3π/4
D. 4π/3
Correct Answer: C
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Functions
Benchmark Number / MA.912.T.1.2
Benchmark / Define and determine sine and cosine using the unit circle.
Also Assesses or Assessed by / MA.912.T.1.1
Item Types / MC
Benchmark Clarification / Missing
Content Limits / *ϴ may be stated in terms of degrees or radians
*ϴ can be determinedthrough multiple rotations around the unit circle
Stimulus Attributes / Items may be set in real world or mathematical context
Response Attributes / Missing
Sample Item / Missing
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Functions
Benchmark Number / MA.912.T.1.3
Benchmark / State and use exact values of trigonometric functions for special angles, i.e. multiples of and (degree and radian measures)
Also Assesses or Assessed by / MA.912.T.1.1
MA.912.T.1.2
Item Types / MC
FR
Benchmark Clarification / Determine the value of the 6 trigonometric functions in terms of degrees with inmultiples of 30, 45, 60,90, and 180 degreesor radians with in multiples of π/6, π /4, π/3, π/2, and π.
Content Limits / Content is limited to determining the value of the 6 trigonometric functions in terms of degrees with inmultiples of 30, 45, 60,90, and 180 degreesor radians with in multiples of π/6, π /4, π/3, π/2, and π.
Stimulus Attributes / Item should be set in mathematical context
Response Attributes / None specified.
Sample Item / Given that trigonometric functions are periodic, what is the exact value of cos (10π/3)?
No Answer!
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Functions
Benchmark Number / MA.912.T.1.4
Benchmark / Find approximate values of trigonometric and inverse trigonometric functions using appropriate technology.
Also Assesses or Assessed by / MA.912.T.2.2
Item Types / MC
FR
Benchmark Clarification / Determine the value of the 6 trigonometric and inverse trigonometric functions with any appropriate domain.
Content Limits / Angles must be in degrees or radians with two or less decimal places.
Stimulus Attributes / Items may be set in real world or mathematical context
Response Attributes / Angles may be in degrees or radians. Output from the inverse functions will be an exact or rounded rational number.
Sample Item / Find the inverse tangent of 1.51. Is this number an angle or the ratio of the sides of the triangle?
*A. 56 and angle
B. 56 and ratio of sides
C. 0.02 and angle
D. 0.02 and ratio of sides
Correct Answer: A
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Functions
Benchmark Number / MA.912.T.1.5:
Benchmark / Make connections between right triangle ratios, trigonometric functions, and circular functions.
Also Assesses or Assessed by / MA.912.T.1.2
Item Types / MC
FR/GR
Benchmark Clarification / Students solve for missing sides or angles of right triangles, given either one non-right angle and one side, or simply two sides. Canrequire knowledge ofthe side ratios of the special right triangles:30-60-90, 45-45-90.Students may also be asked to find equivalent values of the six trigonometric functions for different values of an angle.
Content Limits / Angles must be in degrees with two or less decimal places.
Stimulus Attributes / Items can be set in real-world or numerical contexts, with or without graphics.
Response Attributes / None specified.
Sample Item / Given a 50º angle of a right triangle with a hypotenuse of length 14. Find the exact value of the longer leg of the triangle.
Correct Answer: 14sin(50º) or 14cos(40º)
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Functions
Benchmark Number / MA.912.T.1.6:
Benchmark / Define and graph trigonometric functions using domain, range, intercepts, period, amplitude, phase shift, vertical shift, and asymptotes with and without the use of graphing technology.
Also Assesses or Assessed by / N/A
Item Types / MC
FR
GR
Benchmark Clarification / Students identify the domain, range, intercepts, period, amplitude, transformations, and asymptotes of trigonometric functions or their graphs.
Content Limits / None specified.
Stimulus Attributes / Items should be set in numerical contexts with or without graphics.
Response Attributes / None specified.
Sample Item / Determine the range of the function f(x)=5sin(2x-pi/3)+1
No Correct Answer Given!
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Functions
Benchmark Number / MA.912.T.1.8:
Benchmark / Solve real-world problems involving applications of trigonometric functions using graphing technology when appropriate.
Also Assesses or Assessed by / N/A
Item Types / MC
FR
GR
Benchmark Clarification / Students solve problems based on trigonometric functions or their graphs.
Content Limits / None specified.
Stimulus Attributes / Items should be set in real-world contexts with or without graphics.
Response Attributes / None specified.
Sample Item / The number of hours of daylight varies through the year in any location. A graph of the number of hours of daylight throughout the year is in the form of a sine wave. In a certain location the longest day of 14 hours is on Day 175 and the shortest day of 10 hours is on Day 355. Identify a possible equation for this function.
No Correct Answer Given!
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometry in Triangles
Benchmark Number / MA.912.T.2.1
Benchmark / Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles.
Also Assesses or Assessed by / N/A
Item Types / This benchmark will be assessed using MC items.
Benchmark Clarification / Students will determine the trigonometric function (sine, cosine, tangent, cosecant, secant, or cotangent) using the side lengths of a given right triangle, or students will determine the angle measures of a givenright triangle using inverse trigonometric functions.
Content Limits / Angle measures will be in degrees.
Items may include the expectation of using special right triangle measures or the Pythagorean Theorem.
Items may require multiple steps.
Items may require theuse of calculators to find angle measures.
Stimulus Attributes / Items must be set in mathematical contexts.
Graphics should be used in all items.
Any radical expressions must be in simplified or rationalized form.
Response Attributes / Angle measures will be in degrees.
The trigonometric ratios will be presented in fraction form.
Any radical expressions will be in simplified or rationalized form.
Sample Item / What is the value of sec θ?
A. 8/15
B. 15/17
C. 17/8
*D. 17/15
Correct Answer: D
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometry in Triangles
Benchmark Number / MA.912.T.2.2
Benchmark / Solve real-world problems involving right triangles using technology when appropriate.
Also Assesses or Assessed by / N/A
Item Types / This benchmark will be assessed using MC andFRitems.
Benchmark Clarification / Students will solve real-world problems involving right triangles using calculators as needed.
Students will be required to provide a length or an angle measure.
Students may be asked to solve problems involving angles of elevation, angles of depression, bearings, or other types of real-world problems.
Content Limits / Angle measures will be in degrees.
Items may require multiple steps.
Items may require the use of calculators to find lengths and angle measures.
Stimulus Attributes / Items must be set in real-world contexts.
Graphics may be given to enhance the item, or students may be expected to make a sketch to assist in giving a response.
Items will specify the nature of the response, if the response is not an integer.
Response Attributes / Angle measures will be in degrees.
Multiple-choice and fill-in responses will be in decimal form.
Sample Item / Sample Item: Fill-In Response
Students are building an additional wheelchair ramp to the high school's auditorium. What is the angle of elevation of the fifty-foot ramp, to the nearest tenth of a degree, if the final height of the ramp will be 4 feet?
Correct Answer: 4.6
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometry in Triangles
Benchmark Number / MA.912.T.2.3
Benchmark / Apply the laws of sine and cosine to solve real-world problems using technology.
Also Assesses or Assessed by / N/A
Item Types / This benchmark will be assessed using MC items.
Benchmark Clarification / Students will solve real-world problems involving oblique triangles by applying the Law of Sines or the Law of Cosines which will be provided on the Trigonometry Reference Sheet.
Students may be required to provide a length or an angle measure.
Students may be required to find side lengths before using the Law of Sines or Law of Cosines to solve the real-world problems.
Content Limits / Angles measures will be in degrees.
Items may require multiple steps.
Items may require the use of calculators to find lengths and angle measures.
Stimulus Attributes / Items must be set in real-world contexts.
Graphics may be given to enhance the item, or students may be expected to make a sketch to assist in giving a response.
Items will specify the nature of the response, if the response is not an integer.
Response Attributes / Angle measures will be in degrees.
Multiple-choice responses will be in decimal form.
Sample Item / Two planes leave an airport on different runways at the same time. The runways intersect at an included angle of 100°. One plane travels at 350 miles per hour on a straight flight path, and the other plane travels at 425 miles per hour. How far apart, to the nearest mile, are the planes after 3 hours?
A. 225 miles
B. 684 miles
C. 1504 miles
*D. 1787 miles
Correct Answer: D
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometry in Triangles
Benchmark Number / MA.912.T.2.4
Benchmark / Use the area of triangles given two sides and an angle or three sides to solve real-world problems.
Also Assesses or Assessed by / N/A
Item Types / This benchmark will be assessed using MC and FR items.
Benchmark Clarification / Students will solve real-world problems by finding the area of a triangle by using Heron's Formula, the area of a triangle formula using the sine function, the basic area of a triangle formula, or other means using trigonometric functions when given two sides and an angle or three sides of a triangle.
Heron's Formula and the area of a triangle formula using the sine function will be provided on the Trigonometry Reference Sheet.
Content Limits / Angle measures will be in degrees.
Items may require multiple steps.
Items will require the use of calculators with trigonometric functions.
Stimulus Attributes / Items must be set in real-world contexts.
Graphics may be given to enhance the item, or students may be expected to make a sketch to assist in giving a response.
Items will specify the nature of the response, if the response is not an integer.
Response Attributes / Angle measures will be in degrees.
Multiple-choice and fill-in responses will be in decimal form.
Sample Item / What is the area, to the nearest square foot, of a triangular piece of land that measures 275 feet by 400 feet by 425 feet?
A. 6837 square feet
B. 42,482 square feet
*C. 53,254 square feet
D. 160,351 square feet
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Identities and Equations
Benchmark Number / MA.912.T.3.1
Benchmark / Verify the basic Pythagorean identities, e.g., sin2x + cos2x = 1, and show they are equivalent to the Pythagorean Theorem.
Also Assesses or Assessed by / Missing or N/A
Item Types / Missing
Benchmark Clarification / Missing
Content Limits / Missing
Stimulus Attributes / Missing
Response Attributes / Missing
Sample Item / Missing
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Identities and Equations
Benchmark Number / MA.912.T.3.2
Benchmark / Use basic trigonometric identities to verify other identities and simplify expressions.
Also Assesses or Assessed by / Missing or N/A
Item Types / Missing
Benchmark Clarification / Missing
Content Limits / Missing
Stimulus Attributes / Missing
Response Attributes / Missing
Sample Item / Missing
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Identities and Equations
Benchmark Number / MA.912.T.3.3
Benchmark / Use the sum and difference, half-angle and double-angle formulas for sine, cosine, and tangent, when formulas are provided.
Also Assesses or Assessed by / Missing or N/A
Item Types / Missing
Benchmark Clarification / Missing
Content Limits / Missing
Stimulus Attributes / Missing
Response Attributes / Missing
Sample Item / Missing
Reporting Category / Body of Knowledge / Trigonometry
Standard / Trigonometric Identities and Equations
Benchmark Number / MA.912.T.3.4
Benchmark / Solve trigonometric equations and real-world problems involving applications of trigonometric equations using technology when appropriate.
Also Assesses or Assessed by / Missing or N/A
Item Types / Missing
Benchmark Clarification / Missing
Content Limits / Missing
Stimulus Attributes / Missing
Response Attributes / Missing
Sample Item / Missing
Reporting Category / Body of Knowledge / Trigonometry
Standard / Polar Coordinates and Trigonometric Form of Complex Numbers
Benchmark Number / MA.912.T.4.1
Benchmark / Define polar coordinates and relate polar coordinates to Cartesian coordinates with and without the use of technology.
Also Assesses or Assessed by / N/A
Item Types / This benchmark will be assessed using MC items.
Benchmark Clarification / Students will define polar coordinates.
Students will recognize a graph of a point in polar coordinates in a polar coordinate system.
Students will name polar coordinates by examining a point in a polar coordinate system.
Students will convert polar coordinates to Cartesian coordinates and vice versa with and without calculators.
Content Limits / Items may be solved using calculators that will convert between polar coordinates and Cartesian coordinates.
Items may include points in polar coordinates that have both positive and negative r values.
Angle measures may be in degree or radian measures between -1080° (-6π) and 1080° (6π).
Stimulus Attributes / Items may be set in mathematical or real-world contexts.
Graphics may be given to enhance the item, or students may be expected make a sketch to assist in giving a response.
Response Attributes / Angle measures may be given in degrees or radians.
Points will be listed in ordered pairs.
Sample Item / The polar coordinates of a point are (-4, 270°). Which ordered pair represents the same point in Cartesian coordinates?
A. (-4,0)
*B. (0,4)
C. (-4,4)
D. (-4,-4)
Correct Answer: B
Reporting Category / Body of Knowledge / Trigonometry
Standard / Polar Coordinates and Trigonometric Form of Complex Numbers
Benchmark Number / MA.912.T.4.2
Benchmark / Represent equations given in rectangular coordinates in terms of polar coordinates.
Also Assesses or Assessed by / MA.912.T.1.1, MA.912.T.1.2, MA.912.T.4.1
Item Types / MC
Benchmark Clarification / Convert equations written in rectangular coordinates to equations written in polar coordinates
Content Limits / N/A
Stimulus Attributes / Items must be set in a mathematical context.
Response Attributes / N/A
Sample Item / How do you represent the equation x² + (y - 1)² = 1 in polar form?
Correct Answer A. r = 2 sin θ
Correct Answer B. r² = cos 2θ
Are both answers correct? Or should the correct answer contain both options A and B?
Reporting Category / Body of Knowledge / Trigonometry
Standard / Polar Coordinates and Trigonometric Form of Complex Numbers
Benchmark Number / MA.912.T.4.3
Benchmark / Graph equations in the polar coordinate plane with and without the use of graphing technology.
Also Assesses or Assessed by / N/A
Item Types / This benchmark will be assessed using MC and FR items.
Benchmark Clarification / Students will recognize the equations and graphs in polar coordinates in a polar coordinate system with or without using a graphing utility.
Students will answer questions about the meaning of the constants in polar equations with or without using a graphing utility.
Content Limits / Students may use graphing calculators to view graphs of polar equations.
Equations and graphs will be limited to proper forms of circles, lines, rose curves, lemniscates, or limacons (including cardioids).