Area = ½ bc sinA = ½ ab sinC = ½ ac sinB

Slides:



Advertisements
Similar presentations
Oblique Triangles.
Advertisements

Law of Sines.
The Law of Cosines February 25, 2010.
Law of Sines Law of Cosines
(8 – 2) Law of Sines Learning target: To solve SAA or ASA triangles To solve SSA triangles To solve applied problems We work with any triangles: Important.
Our last new Section…………5.6. Deriving the Law of Cosines a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) a c b AB(c,0) C(x, y) In all three cases: Rewrite:
The Law of Sines and The Law of Cosines
Chapter 6 – Trigonometric Functions: Right Triangle Approach
The Law of Sines and The Law of Cosines
Module 8 Lesson 5 Oblique Triangles Florben G. Mendoza.
Section SOLVING OBLIQUE TRIANGLES
Trigonometry 2 Aims Solve oblique triangles using sin & cos laws Objectives Calculate angles and lengths of oblique triangles. Calculate angles and lengths.
19. Law of Sines. Introduction In this section, we will solve (find all the sides and angles of) oblique triangles – triangles that have no right angles.
Triangles- The Ambiguous Case Lily Yang Solving Triangles If you are given: Side-Side-Side (SSS) or Side-Angle-Side (SAS), use the Law of Cosines.
Copyright © Cengage Learning. All rights reserved. 6 Additional Topics in Trigonometry.
Finding Areas with Trigonometry. Objectives I can use trigonometry to find the area of a triangle.
5.6 Law of Cosines. I. Law of Cosines In any triangle with opposite sides a, b, and c: The Law of Cosines is used to solve any triangle where you are.
13.4 L AW OF S INES 13.5 L AW OF COSINES Algebra II w/ trig.
Lesson 6.1 Law of Sines. Draw any altitude from a vertex and label it k. Set up equivalent trig equations not involving k, using the fact that k is equal.
Quiz 5-5 Solve for the missing angle and sides of Triangle ABC where B = 25º, b = 15, C = 107º Triangle ABC where B = 25º, b = 15, C = 107º 1. A = ? 2.
1 Law of Cosines Digital Lesson. 2 Law of Cosines.
6.2 The Law of Cosines.
Law of Sines AB C a c b h Warm-Up: Find two different expressions for h, one using angle A and one using angle B Objective: Be able to use the law of sines.
LAW OF SINES: THE AMBIGUOUS CASE MENTAL DRILL Identify if the given oblique triangle can be solved using the Law of Sines or the Law of Cosines 1.
Review 1. Solving a right triangle. 2. Given two sides. 3. Given one angle and one side.
Copyright © 2011 Pearson, Inc. 5.5 Law of Sines Goal: Solve triangles that have no solution, one solution, or two solutions.
5.5 Law of Sines. I. Law of Sines In any triangle with opposite sides a, b, and c: AB C b c a The Law of Sines is used to solve any triangle where you.
6.1 Law of Sines. Introduction Objective: Solve oblique triangles To solve: you must know the length of one side and the measures of any two other parts.
If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle.
Trigonometry Section 6.1 Law of Sines. For a triangle, we will label the angles with capital letters A, B, C, and the sides with lowercase a, b, c where.
Notes Over 8.1 Solving Oblique Triangles To solve an oblique triangle, you need to be given one side, and at least two other parts (sides or angles).
Chapter 6.  Use the law of sines to solve triangles.
Warm up   A 5.2 m ladder leans against a wall. The bottom of the ladder is 1.9 m from the wall. What angle does the ladder make with the ground (to.
Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and.
The Law of Cosines. Let's consider types of triangles with the three pieces of information shown below. SAS You may have a side, an angle, and then another.
Oblique Triangles Oblique Triangle – a non-right triangle. Oblique Triangle – a non-right triangle. It may be acute. It may be obtuse.
Lesson 6.5 Law of Cosines. Solving a Triangle using Law of Sines 2 The Law of Sines was good for: ASA- two angles and the included side AAS- two angles.
Lesson 6.1- Law of Sines Provided by Vivian Skumpija and Amy Gimpel.
1 What you will learn  How to solve triangles by using the Law of Cosines  How to find the area of triangles if the measures of the three sides are given.
Section 4.2 – The Law of Sines. If none of the angles of a triangle is a right angle, the triangle is called oblique. An oblique triangle has either three.
Copyright © Cengage Learning. All rights reserved. 6 Additional Topics in Trigonometry.
Toothpicks and PowerPoint Ambiguous Case of the Law of Sines Pam Burke Potosi High School #1 Trojan Drive Potosi, MO
Notes Over 8.2 Solving Oblique Triangles To solve an oblique triangle, you need to be given one side, and at least two other parts (sides or angles).
6.2 Law of Cosines *Be able to solve for a missing side or angle using law of cosines.
Math 20-1 Chapter 2 Trigonometry 2.4 The Cosine Law Teacher Notes.
Law of Cosines. h a c A B C x D b - x b To derive the formula, fine the relationship between a, b, c, and A in this triangle. a 2 = (b – x) 2 + h 2 a.
a = 6, b = 4, C = 60 º 6 Sin A = 4 Sin B = c Sin 60º.
Law of Sines Use it when you are given Angle-Angle-Side (AAS) Angle-Side-Angle (ASA) Side-Side-Angle (SSA)
Law of Sines.
Law of Cosines. SAS Area Formula: A b c Heron’s SSS Area Formula: b c a.
Make it a great day!! The choice is yours!! Complete the Do Now.
Welcome to Week 5 College Trigonometry. Secant Secant with a graphing calculator.
Additional Topics in Trigonometry
Oblique Triangles.
Law of Sines.
Oblique Triangles.
Lesson: _____ Section 6.1 The Law of Sines
Lesson 37 continued Get out your notes from yesterday.
Law of Sines.
Theorem The area A of a triangle is
Essential question: How do I solve oblique triangles?
19. Law of Sines.
Law of Sines What You will learn:
50 a 28.1o Warm-up: Find the altitude of the triangle.
Section 6.1.
Law of Sines and Law of Cosines
Oblique Triangles.
Law of Sines. Law of Sines Non Right Triangles How is a triangle classified if none of the angles are 90? Oblique Labeling Oblique Triangles To solve.
7.1, 7.2, 7.3 Law of Sines and Law of Cosines
Chapter 2 Trigonometry 2.4 The Cosine Law
Presentation transcript:

Area = ½ bc sinA = ½ ab sinC = ½ ac sinB The area of an oblique triangle can be found by using the following 
formula: Area = ½ bc sinA  =  ½ ab sinC  =  ½ ac sinB Since not all triangles are labeled A,B,C, you should remember 
that the AREA of a triangle = 1/2 the product of _________and the sine of ___________.

Area = ½ bc sinA = ½ ab sinC = ½ ac sinB Find the area of each triangle described below. 1) A = 36o, b = 3 feet, c = 5 feet Area = ½ bc sinA = ½ ab sinC = ½ ac sinB

Area = ½ bc sinA = ½ ab sinC = ½ ac sinB 2) C = 100o, a = 12 inches, b = 10 inches Area = ½ bc sinA = ½ ab sinC = ½ ac sinB

Continuing our discussion of dealing with oblique (non­right)
 triangles, today we will look at the Law of Cosines.   Yesterday we talked finished talking about Law of Sines. (Used 
with: AAS ASA SSA *check for ambiguous cases! - 0,1, or 2 Triangles Law of Cosines is used for: SAS SSS when there isn't an angle and opposite side pair

a2 =b2+c2 -2bc cosA b2 =a2+c2 -2ac cosB c2 =a2+b2 -2ab cosC Law of Cosines a2 =b2+c2 -2bc cosA b2 =a2+c2 -2ac cosB c2 =a2+b2 -2ab cosC

a2 =b2+c2 -2bc cosA b2 =a2+c2 -2ac cosB c2 =a2+b2 -2ab cosC Use the law of cosines to find the missing sides and angles. C a2 =b2+c2 -2bc cosA b2 =a2+c2 -2ac cosB c2 =a2+b2 -2ab cosC 8 a 700 B 6 A

a2 =b2+c2 -2bc cosA b2 =a2+c2 -2ac cosB c2 =a2+b2 -2ab cosC Use the law of cosines to find the missing sides and angles. C a2 =b2+c2 -2bc cosA b2 =a2+c2 -2ac cosB c2 =a2+b2 -2ab cosC 7 8 A 12 B

Area = 1/4 √ p(p-2a)(p-2b)(p-2c) Another method for finding the area of a triangle  (if we do not know the height) is Heron's Formula,
 which is based on the perimeter, p, of the triangle,
 p = a + b + c. Area = 1/4 √ p(p-2a)(p-2b)(p-2c)

use law of cosines to find an a Recap! Draw and label triangle. Decide which law to use base on similarity theorem. AAS ASA SSA SAS SSS check ambiguous cases for how many triangles use law of cosines to find an a missing piece use law of sines to find missing measures use law of sines to find missing measures Find Area