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Standard 4 : Apply trigonometry to general triangles This document was generated on CPALMS - www.cpalms.org Number: MAFS.912.G-SRT.4 Title: Apply trigonometry to general triangles Type: Cluster Subject: Mathematics Grade: 912 Domain: Geometry: Similarity, Right Triangles, & Trigonometry Related Standards Code MAFS.912.G-SRT.4.9: MAFS.912.G-SRT.4.10: MAFS.912.G-SRT.4.11: Description Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Prove the Laws of Sines and Cosines and use them to solve problems. Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). Related Resources Perspectives Video: Professional/Enthusiast Name Description Celestial Navigation and Trigonometry: Lost at sea? We can use a sextant, the stars, and trigonometry to find our location! Lesson Plan Name Description Students will be introduced to a derivation of the Deriving and Applying the Law of Sines: Law of Sines and apply the Law of Sines to solve triangles. This lesson introduces the law of sine. It is designed to give students practice in using the law Following the Law of Sine: to guide understanding. The summative assessment requires students to use the law of sine to plan a city project. This lesson unit is intended to help you assess how well students are able to use geometric properties to solve problems. In particular, the lesson will help you identify and help students who have the following difficulties solving problems by determining the lengths of the sides in right Geometry Problems: Circles and Triangles: triangles and finding the measurements of shapes by decomposing complex shapes into simpler ones. The lesson unit will also help students to recognize that there may be different approaches to geometrical problems, and to understand the relative strengths and weaknesses of those approaches. In this lesson, students determine when to use the Law of Cosine and/or the Law of Sine to calculate The Laws of Sine and Cosine: the measures of sides and angles of non-right triangles. 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. Will You Survive?: Note: This is not an introductory lesson for the standard. Educational Game Name Law of Cosines Game 2: Description Play a game using the law of cosines to find the measure of angles in non-right triangles or the cosine of an angle. Answer questions with and without a calculator. Select Teach Me to Law of Sines Game: Law of Sines Game 2: learn how to set up equations with trig ratios and solve for the measure of angles prior to beginning play. Hints and feedback are available as you play. Play a game using the law of sines to solve for the length of sides in non-right triangles or use trig ratios to set up an equation. Answer questions with and without a calculator. Students may select Teach Me to learn how to set up equations with trig ratios and solve for the length of sides prior to beginning play. Hints and feedback are provided to players. Play a game using the law of sines to find the measure of angles in non-right triangles or the sine of an angle. Answer questions with and without a calculator. Students may select Teach Me to learn how to set up equations with trig ratios and solve for the measure of angles prior to beginning play. Hints and feedback are provided to players. Video/Audio/Animation Name Description "This video lesson presents a real world problem that can be solved by using the Pythagorean theorem. The problem faces a juice seller daily. He has equilateral barrels with equal heights and he always tries to empty the juice of two barrels into a third barrel that has a volume equal to the sum of the volumes of the two barrels. This juice seller wants to find a simple way to help him select the right barrel without wasting time, and without any calculations - since he is ignorant of mathematics. The MIT BLOSSOMS - The Juice Seller’s Problem: prerequisite for this lesson includes knowledge of the following: the Pythagorean theorem; calculation of a triangle's area knowing the angle between its two sides; cosine rule; calculation of a circle's area; and calculation of the areas and volumes of solids with regular bases. Materials necessary include: equilateral containers of equal heights; sand; and measuring devices. Examples of in-class activities for the breaks between video segments include class discussions, individual calculations and small group problem solving." (from MIT Blossoms' "Pythagoras and the Juice Seller") Virtual Manipulative Name Right Triangle Calculator: Triangle Solver: Description "Free online calculator will calculate the side lengths and angles for any right triangle. Just input any valid combination of sides and/or angles, and let the calculator do the rest! This free tool also draws a downloadable image of your triangle." (from Math Worksheets Go!) The triangle solver manipulative displays a triangle with some of its sides and angles given. The students are then asked to determine values of the remaining sides and angles. Students are motivated to choose a workable strategy such as using the Pythagorean theorem, the sine, cosine, tangent relationships, the law of sines, or the law of cosines. They are directed through the key steps of the chosen strategy to find the unknown sides and the angles. Problem-Solving Task Name Seven Circles III: Description 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. Tutorial Name The basics of trigonometry: Description This tutorial shows the basics of trigonometry, sine, cosine, and tangent. Students will determine oppopsite sides, adjacent sides and the hypotenuse of right triangles. Student Resources Title Description Law of Cosines Game 2: Law of Sines Game: Law of Sines Game 2: Seven Circles III: The basics of trigonometry: Triangle Solver: Play a game using the law of cosines to find the measure of angles in non-right triangles or the cosine of an angle. Answer questions with and without a calculator. Select Teach Me to learn how to set up equations with trig ratios and solve for the measure of angles prior to beginning play. Hints and feedback are available as you play. Play a game using the law of sines to solve for the length of sides in non-right triangles or use trig ratios to set up an equation. Answer questions with and without a calculator. Students may select Teach Me to learn how to set up equations with trig ratios and solve for the length of sides prior to beginning play. Hints and feedback are provided to players. Play a game using the law of sines to find the measure of angles in non-right triangles or the sine of an angle. Answer questions with and without a calculator. Students may select Teach Me to learn how to set up equations with trig ratios and solve for the measure of angles prior to beginning play. Hints and feedback are provided to players. 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. This tutorial shows the basics of trigonometry, sine, cosine, and tangent. Students will determine oppopsite sides, adjacent sides and the hypotenuse of right triangles. The triangle solver manipulative displays a triangle with some of its sides and angles given. The students are then asked to determine values of the remaining sides and angles. Students are motivated to choose a workable strategy such as using the Pythagorean theorem, the sine, cosine, tangent relationships, the law of sines, or the law of cosines. They are directed through the key steps of the chosen strategy to find the unknown sides and the angles. Parent Resources Title Seven Circles III: Description 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. Triangle Solver: The triangle solver manipulative displays a triangle with some of its sides and angles given. The students are then asked to determine values of the remaining sides and angles. Students are motivated to choose a workable strategy such as using the Pythagorean theorem, the sine, cosine, tangent relationships, the law of sines, or the law of cosines. They are directed through the key steps of the chosen strategy to find the unknown sides and the angles.