MA.912.T.1.3

Apply the Law of Sines and the Law of Cosines to solve mathematical and real-world problems involving triangles.
General Information
Subject Area: Mathematics (B.E.S.T.)
Grade: 912
Strand: Trigonometry
Status: State Board Approved

Benchmark Instructional Guide

Connecting Benchmarks/Horizontal Alignment

 

Terms from the K-12 Glossary

  • Angle 
  • Triangle 

 

Vertical Alignment

Previous Benchmarks

Next Benchmarks

 

Purpose and Instructional Strategies

In grade 8, students solved problems involving triangles using Pythagorean Theorem, Triangle Sum Theorem and Triangle Inequality Theorem. In Geometry, students apply the Law of Sines and the Law of Cosines to solve problems involving right and non-right triangles. In later courses, students will use other trigonometric relationships and identities to solve real-world and mathematical problems involving sides and angles of triangles. Additionally, students will use knowledge of the Law of Sines and Law of Cosines when learning about vectors.
  • 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., θ, α, β, γ).
  • The Law of Sines states that given a triangle with side lengths a, b and c, and angle measures α, β and γ (opposite to the sides respectively), then asin α = bsin β = csin γ or sin αa = sin βb = sin γc.
  • Instruction includes making the connection between the Law of Sines and the congruence criteria of Angle-Side-Angle and Angle-Angle-Side. 
    • For example, when given two angle measures and the included side length, students can use the Law of Sines to determine the missing side lengths and angle measures and justify that there is one unique triangle as indicated by the Angle-Side-Angle congruence criterion. 
  • Instruction includes making the connection between the Law of Sines and the Side-Side-Angle criterion, which is referred to as the ambiguous case. 
    • For example, when given two side lengths and an angle measure of an opposite angle students can use the Law of Sines to determine the value of the sines of the missing angle measures. If the value of either of the sines is greater than one, then no triangle exists with those angle measures. If the value of both of the sines is less than one, then there are two possible triangles, one acute and one obtuse. If the value of either of the sines is equal to one, then there is exactly one triangle, a right triangle, with those measures. 
  • The Law of Cosines states that given a triangle with side lengths a, b and c, and angle measures α, β and γ (opposite to the sides respectively), then a2 = b2 + c2 − 2bc cos α, b2 = a2 + c2 − 2ac cos β and c2 = a2 + b2 − 2ab cos γ. 
  • Instruction includes making the connection between the Law of Cosines and the congruence criteria of Side-Side-Side and Side-Angle-Side. 
    • For example, when given three side lengths, students can use the three versions of the Law of Cosines to determine the missing angle measures and justify that there is one unique triangle as indicated by the Side-Side-Side congruence criterion. However, if the side lengths do not satisfy the Triangle Inequality Theorem, then no triangle exists. 
  • Instruction includes making the connection between the Law of Cosines and the Side-Side-Angle criterion, which is referred to as the ambiguous case. 
    • For example, when given two side lengths and an angle measure of an opposite angle students can use one of the versions of the Law of Cosines to obtain a quadratic equation for the missing side length. If there is no real solutions, then no triangle exists. If there are two real solutions, then there are two possible triangles, one acute and one obtuse. If there is exactly one real solution, then there is exactly one triangle, a right triangle. 
  • Instruction includes the understanding that the Pythagorean Theorem is a special case of the Law of Cosines when the included angle measure is 90°, whose cosine is zero. (MTR.5.1) 
  • Instruction includes the understanding that sines and cosines of obtuse angles are necessary when solving problems involving obtuse triangles. Enrichment of this benchmark includes the connection to the unit circle.

 

Common Misconceptions or Errors

  • Students may have trouble keeping track of which sides or angles belong in the Laws of Sines and Cosines.

 

Instructional Tasks

Instructional Task 1 (MTR.3.1, MTR.4.1)
  • Triangle ABC is an acute triangle. The measure of AB is 17.8 inches, the measure of BC is 16.5 inches and the measure of angle C is 75°.
    Triangle
    • Part A. Determine the measure of angle A. Compare your method used with a partner.
    • Part B. Determine the measure of angle B.

Instructional Task 2 (MTR.4.1, MTR.5.1)
  • Part A. Given side lengths 10, 11 and 22, use the Law of Cosines to determine if a triangle exists with those side lengths.
  • Part B. How could you determine, without using the Law of Cosines, whether such a triangle exists?
  • Part C. If the triangle from Part A does not exist, change one of its side lengths so that a triangle does exist.

 

 

Instructional Items

Instructional Item 1
  • Find the remaining measures of the angles and sides of triangle ABC.
Triangle

 

*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.

Related Courses

This benchmark is part of these courses.
1202340: Precalculus Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1206320: Geometry Honors (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Lesson Plans

Deriving and Applying the Law of Sines:

Students will be introduced to a derivation of the Law of Sines and apply the Law of Sines to solve triangles.

Type: Lesson Plan

The Laws of Sine and Cosine:

In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate the measures of sides and angles of non-right triangles.

Type: Lesson Plan

Will You Survive?:

Students are stranded on a desert island and will need to use the law of sines in order to find the quickest path to a rescue vessel.

Note: This is not an introductory lesson for the standard.

Type: Lesson Plan

Following the Law of Sine:

This lesson introduces the law of sine. It is designed to give students practice in using the law to guide understanding. The summative assessment requires students to use the law of sine to plan a city project.

Type: Lesson Plan

Perspectives Video: Professional/Enthusiast

Celestial Navigation and Trigonometry:

<p>Lost at sea? We can use a sextant, the stars, and trigonometry to find our location!</p>

Type: Perspectives Video: Professional/Enthusiast

Problem-Solving Task

Seven Circles III:

This provides an opportunity to model a concrete situation with mathematics. Once a representative picture of the situation described in the problem is drawn (the teacher may provide guidance here as necessary), the solution of the task requires an understanding of the definition of the sine function.

Type: Problem-Solving Task

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.