Trigonometry Item Specifications
Reporting
Category /
Body of
Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Discrete Mathematics
Vectors
MA.912.D.9.1
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.
MA. 912.D.9.3
Benchmark
Clarification
MC
FR
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
1
Trigonometry Item Specifications
Reporting
Category / Body
of Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses or
Assessed by
Item Types
Discrete Mathematics
Vectors
MA.912.D.9.2
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.
MA. 912.D.9.3
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
Stimulus
Attributes
Scalars will be rational numbers only.
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
2
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Discrete Mathematics
Vectors
MA.912.D.9.3
Use vectors to model and solve application problems.
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
3
Trigonometry Item Specifications
4
Reporting
Category / Body
of Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses or
Assessed by
Item Types
Trigonometry
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?
Trigonometric Functions
MA.912.T.1.1
Convert between degree and radian measures.
MA.912.T.1.2 MA.912.T.4.1 MA.912.T.4.2
MC
A. 9π
B. π/4
C. 3π/4
D. 4π/3
Correct Answer: C
GR
MA.912.T.4.3
FR
Trigonometry Item Specifications
Reporting
Category / Body
of Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Content Limits
*ϴ may be stated in terms of degrees or radians
*ϴ can be determined through multiple rotations around the unit circle
Stimulus
Attributes
Items may be set in real world or mathematical context
Response
Attributes
Missing
Sample Item
Missing
Trigonometric Functions
MA.912.T.1.2
Define and determine sine and cosine using the unit circle.
MA.912.T.1.1
MC
Missing
5
Trigonometry Item Specifications
Reporting
Category / Body
of Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometric Functions
MA.912.T.1.3
State and use exact values of trigonometric functions for special angles, i.e. multiples
of and (degree and radian measures)
MA.912.T.1.1
MA.912.T.1.2
MC
FR
Determine the value of the 6 trigonometric functions in terms of degrees with
in multiples of 30, 45, 60, 90, and 180 degrees or 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 in multiples of 30, 45, 60, 90, and 180 degrees or 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!
6
Trigonometry Item Specifications
Reporting
Category / Body
of Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses or
Assessed by
Item Types
Trigonometry
Trigonometric Functions
MA.912.T.1.4
Find approximate values of trigonometric and inverse trigonometric functions using
appropriate technology.
MA.912.T.2.2
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
7
Trigonometry Item Specifications
Reporting
Category /
Body of
Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Benchmark
Clarification
Content Limits
Trigonometry
Trigonometric Functions
MA.912.T.1.5:
Make connections between right triangle ratios, trigonometric functions, and circular
functions.
MA.912.T.1.2
MC
FR/GR
Students solve for missing sides or angles of right triangles, given either one non-right
angle and one side, or simply two sides. Can require knowledge of the 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.
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)
8
Trigonometry Item Specifications
Reporting
Category / Body
of Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses or
Assessed by
Item Types
Trigonometry
Trigonometric Functions
MA.912.T.1.6:
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.
N/A
Benchmark
Clarification
MC
FR
GR
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
Sample Item
None specified.
Determine the range of the function f(x)=5sin(2x-pi/3)+1
No Correct Answer Given!
9
Trigonometry Item Specifications
Reporting
Category /
Body of
Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometric Functions
MA.912.T.1.8:
Solve real-world problems involving applications of trigonometric functions using
graphing technology when appropriate.
N/A
MC
FR
GR
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
Sample Item
None specified.
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!
10
Trigonometry Item Specifications
Reporting
Category /
Body of
Knowledge
Standard
Benchmark
Number
Benchmark
Also Assesses
or Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometry in Triangles
MA.912.T.2.1
Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant,
cosecant) in terms of angles of right triangles.
N/A
This benchmark will be assessed using MC items.
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 given right 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 the use 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
11
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometry in Triangles
MA.912.T.2.2
Solve real-world problems involving right triangles using technology when
appropriate.
N/A
This benchmark will be assessed using MC and FR items.
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
12
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
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
Trigonometry in Triangles
MA.912.T.2.3
Apply the laws of sine and cosine to solve real-world problems using technology.
N/A
This benchmark will be assessed using MC items.
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.
Correct Answer: D
13
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometry in Triangles
MA.912.T.2.4
Use the area of triangles given two sides and an angle or three sides to solve realworld problems.
N/A
This benchmark will be assessed using MC and FR items.
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
14
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometric Identities and Equations
MA.912.T.3.1
Verify the basic Pythagorean identities, e.g., sin2x + cos2x = 1, and show they are
equivalent to the Pythagorean Theorem.
Missing or N/A
Missing
Missing
Content Limits
Missing
Stimulus Attributes
Missing
Response Attributes
Missing
Sample Item
Missing
15
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Content Limits
Trigonometry
Trigonometric Identities and Equations
MA.912.T.3.2
Use basic trigonometric identities to verify other identities and simplify
expressions.
Missing or N/A
Missing
Missing
Missing
Stimulus Attributes
Response Attributes
Missing
Missing
Sample Item
Missing
16
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometric Identities and Equations
MA.912.T.3.3
Use the sum and difference, half-angle and double-angle formulas for sine, cosine,
and tangent, when formulas are provided.
Missing or N/A
Missing
Missing
Content Limits
Missing
Stimulus Attributes
Missing
Response Attributes
Missing
Sample Item
Missing
17
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Trigonometric Identities and Equations
MA.912.T.3.4
Solve trigonometric equations and real-world problems involving applications of
trigonometric equations using technology when appropriate.
Missing or N/A
Missing
Missing
Content Limits
Missing
Stimulus Attributes
Missing
Response Attributes
Sample Item
Missing
Missing
18
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Trigonometry
Polar Coordinates and Trigonometric Form of Complex Numbers
MA.912.T.4.1
Define polar coordinates and relate polar coordinates to Cartesian coordinates with
and without the use of technology.
N/A
This benchmark will be assessed using MC items.
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.
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)
Sample Item
Correct Answer: B
19
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Content Limits
Stimulus Attributes
Trigonometry
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
Polar Coordinates and Trigonometric Form of Complex Numbers
MA.912.T.4.2
Represent equations given in rectangular coordinates in terms of polar coordinates.
MA.912.T.1.1, MA.912.T.1.2, MA.912.T.4.1
MC
Convert equations written in rectangular coordinates to equations written in polar
coordinates
N/A
Items must be set in a mathematical context.
Are both answers correct? Or should the correct answer contain both options A
and B?
20
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Content Limits
Trigonometry
Polar Coordinates and Trigonometric Form of Complex Numbers
MA.912.T.4.3
Graph equations in the polar coordinate plane with and without the use of graphing
technology.
N/A
This benchmark will be assessed using MC and FR items.
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.
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).
Stimulus Attributes
Items must be set in a mathematical context.
Polar equations will be given in proper form.
Students will match a polar equation with its graph or answer a question about a
polar equation or its graph.
Graphs will be shown in radian measures only.
Response Attributes
Polar equations will be given in proper form.
Graphs will be shown in radian measures only.
Sample Item
Which of the folllowing describes the graph of r = 3 + 4sin θ?
*A. a limaçon with an inner loop
B. a limaçon with no inner loop
C. a lemniscate symmetric with respect to the polar axis, θ = π/2 , and the pole
D. a rose curve with 4 petals
Correct Answer: A
21
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Content Limits
Stimulus Attributes
Trigonometry
Polar Coordinates and Trigonometric Form of Complex Numbers
MA.912.T.4.4
Define the trigonometric form of complex numbers, convert complex numbers to
trigonometric form, and multiply complex numbers in trigonometric form.
N/A
This benchmark will be assessed using MC and FR items.
Students will convert complex numbers from standard form to trigonometric form
and vice-versa.
Students will multiply complex numbers in trigonometric form.
Angle measures may be in degree or radian measures between 0° (0π) and 360°
(2π).
Items must be set in a mathematical context.
Items involving the conversion of complex numbers must use angles found on the
unit circle.
Response Attributes
Items involving the multiplication of complex numbers in trigonometric form will
have responses in trigonometric form.
Responses will be exact values.
Sample Item
What is the product of 12(cos 45° + i sin 45°) and 6(cos 15° + i sin 15°)?
A. 2(cos 15° + i sin 15°)
B. 2(cos 3° + i sin 3°)
*C. 72(cos 60° + i sin 60°)
D. 72(cos 675° + i sin 675°)
Correct Answer: C
22
Trigonometry Item Specifications
Reporting Category /
Body of Knowledge
Standard
Benchmark Number
Benchmark
Also Assesses or
Assessed by
Item Types
Benchmark
Clarification
Content Limits
Stimulus Attributes
Trigonometry
Polar Coordinates and Trigonometric Form of Complex Numbers
MA.912.T.4.5
Apply DeMoivre's Theorem to perform operations with complex numbers.
N/A
This benchmark will be assessed using MC and FR items.
Students will find powers of complex numbers written in rectangular form or in
trigonometric form by applying DeMoivre's Theorem.
Angle measures may be in degree or radian measures between 0° (0π) and 360°
(2π).
Items must be set in a mathematical context.
Items will only include problems whose solutions include angles found on the unit
circle.
Complex numbers may be written in either rectangular form or trigonometric form.
Response Attributes
Complex numbers may be written in rectangular or trigonometric form.
Complex numbers will be written in standard form.
Responses will be exact values.
Sample Item
Sample Item: Fill-In Response
What is (1 + i) raised to the fourth power?
Sample Correct Answer: 4
23