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Clarifications
Clarification 1: Instruction includes procedural fluency with the relationships of side lengths in special right triangles having angle measures of 30°-60°-90° and 45°-45°-90°.Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
- Angle
- Equilateral Triangle
- Hypotenuse
- Isosceles Triangle
- Right Triangle
Vertical Alignment
Previous Benchmarks
Next Benchmarks
Purpose and Instructional Strategies
In grade 8, students solved problems involving right triangles using the Pythagorean Theorem. In Geometry, students use their understanding of sine, cosine and tangent to solve mathematical and real-world problems involving right triangles. In later courses, students will extend this knowledge to solve more difficult problems with right triangles, and extend the concept of trigonometric ratios to trigonometric functions on the unit circle and the number line.- Within the Geometry course, the expectation is to use angle measures given in degrees and not in radians. Additionally, it is not the expectation for students to master the trigonometric ratios of secant, cosecant and cotangent within this course.
- It is customary to use Greek letters to represent angle measures (e.g., Ø, α, β, γ).
- Problem types include cases where some information about the side lengths or angle measures of a right triangle is missing and one must use trigonometric ratios, inverse of trigonometric ratios or Pythagorean Theorem to determine the unknown length(s) or angle measure(s) within a mathematical or real-world context.
- Instruction includes the concept of inverse trigonometric ratios to determine unknown angle measures and how to find these values using technology, including a calculator. Students should have practice using both notations for the inverse trigonometric ratios (sin−1 A or arcsin A; cos−1 A or arccos A; and tan−1 A or arctan A).
- Instruction includes exploring the relationships of the side lengths of special right triangles 45° − 45° − 90° and 30° − 60° − 90°.
- For example, students should realize that the special right triangle 45° − 45° − 90° is an isosceles right triangle. Therefore, two of its angle measures and side lengths are equivalent. So, if a side length is units, then students can use the Pythagorean Theorem to determine that the hypotenuse is units. Additionally, students can make the connection to its trigonometric ratios: sin 45° = (or equivalently ); cos 45° = (or equivalently ); and tan 45° = 1.
- For example, students should realize that the special right triangle 30° − 60° − 90° is half of an equilateral triangle. Students can use that knowledge to determine that the shorter leg is one-half the length of the hypotenuse. So, if the shorter leg is units and the hypotenuse is 2 units, then students can use the Pythagorean Theorem to determine that the other leg is units. Additionally, students can make the connection to its trigonometric ratios such as, sin 30° = ; cos 30° = and tan 30° = (or equivalently ).
Common Misconceptions or Errors
- Students may choose the incorrect trigonometric ratio when solving problems.
- Students may misidentify the sides of triangles.
- For example, students may identify the hypotenuse as being the adjacent leg or confuse the adjacent and opposite sides.
Instructional Tasks
Instructional Task 1 (MTR.3.1)- ABCD is a square.
- Part A. What is the measure of segment BD?
- Part B. What is the measure of segment AC?
- Part C. If the measure of segment BD is 14 units, what is the measure of segment BC?
Instructional Task 2 (MTR.7.1)
- Part A. A company is requesting equilateral tiles to be made for their new office floor. If the height of the tile is approximately 10.4 inches, what is the length of the sides of the triangle?
- Part B. The same company decides they also want to use half of a square with the side the same length as the height of the equilateral triangle. What is the length of the hypotenuse of the triangle formed from taking half of the square?
Instructional Items
Instructional Item 1- The logo of a local construction company contains an equilateral triangle. The height of the triangle is 10 units. What is the length of the measure of each side of the triangle?
Instructional Item 2
- The right triangle ABC is shown. Angle B is the right angle and the length of AB is 1.5 centimeters and the length of BC is 3.1 centimeters.
- Part A. Determine the measure of angles A and C.
- Part B. Determine the length of AC.
Related Courses
Related Access Points
Related Resources
Formative Assessments
Lesson Plans
Original Student Tutorial
Perspectives Video: Experts
Presentation/Slideshow
Problem-Solving Tasks
MFAS Formative Assessments
Students are asked to solve a problem in a real world context requiring the use of the Pythagorean Theorem.
Students are asked to use 30-60-90 triangle relationships to observe and explain the relationship between sin 30 and cos 60 (or sin 60 and cos 30).
Students are asked to find an unknown length in a real world context requiring right triangle trigonometry.
Students are asked to determine the length of a side of a right triangle in a real-world problem.
Students are asked to find an unknown length in a real world context requiring right triangle trigonometry.
Students are asked to explain the relationship between sine and cosine of the acute angles of a right triangle.
Students are asked to explain the relationship among angles in a diagram involving a right triangle and to find one angle of the right triangle.
Students are asked to solve a problem in a real world context requiring the use of the Pythagorean Theorem.
Students are asked to find the angle of elevation in a real world situation modeled by a right triangle.
Students are asked to solve a problem in a real world context using the Pythagorean Theorem.
Original Student Tutorials Mathematics - Grades 9-12
Learn how to use trigonometric ratios to find the heights of famous monuments and solve a real-world application in this interactive tutorial.
Student Resources
Original Student Tutorial
Learn how to use trigonometric ratios to find the heights of famous monuments and solve a real-world application in this interactive tutorial.
Type: Original Student Tutorial
Problem-Solving Tasks
Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)
Type: Problem-Solving Task
This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.
Type: Problem-Solving Task
This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.
Type: Problem-Solving Task
This task engages students in an open-ended modeling task that uses similarity of right triangles.
Type: Problem-Solving Task
This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.
Type: Problem-Solving Task
This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.
Type: Problem-Solving Task
In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.
Type: Problem-Solving Task
Parent Resources
Problem-Solving Tasks
Using a chart of diameters of different denominations of coins, students are asked to figure out how many coins fit around a central coin. (For this task, United States coins are used, but the task can be adapted for coins from other countries.)
Type: Problem-Solving Task
This problem solving task asks students to find the area of an equilateral triangle. Various solutions are presented that include the Pythagoren theorem and trigonometric functions.
Type: Problem-Solving Task
This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.
Type: Problem-Solving Task
This task engages students in an open-ended modeling task that uses similarity of right triangles.
Type: Problem-Solving Task
This problem solving task gives an interesting context for implementing ideas from geometry and trigonometry.
Type: Problem-Solving Task
This task applies geometric concepts, namely properties of tangents to circles and of right triangles, in a modeling situation. The key geometric point in this task is to recognize that the line of sight from the mountain top towards the horizon is tangent to the earth. We can then use a right triangle where one leg is tangent to a circle and the other leg is the radius of the circle to investigate this situation.
Type: Problem-Solving Task
In this task, students use trigonometric functions to model the movement of a point around a wheel and, through space. Students also interpret features of graphs in terms of the given real-world context.
Type: Problem-Solving Task
