MODULE 5

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writers: Michelle Catalan, Ma. Aylene C. Fabillar
• Cover Illustrator: Joel J. Estudillo
• MATHEMATICS Quarter 4: Module 5 LAWS OF SINES AND COSINES
• Department of Education National Capital Region SCHOOLS DIVISION OFFICE MARIKINA CITY
• What I Need to Know


  illustrates laws of sines and cosines
• What I Know
• Encircle the letter that corresponds to the correct answer.
• Laws of Sines

  In any oblique triangle ABC with sides a, b and c, a/sin A = b/sin B = c/sin C or sin A/a = sin B/b = sin C/c
• Case 1: Given two angles and an opposite side (SAA)


  1. Illustrate the given triangle, find the value of b

  2. Use the Laws of Sines: a/sin A = b/sin B

  3. Solve for b
• Case 2: Given two angles and one side (ASA)


  1. Illustrate the given triangle, identify the value of c

  2. A+B+C = 180°, find A

  3. Use the Laws of Sines: a/sin A = c/sin C

  4. Solve for c
• Case 3: Given two sides and an opposite angle (SSA)


  1. Use the Law of Sines to solve for the value of ∠B

  2. Solve for ∠C and side c
• Oblique triangle


  A triangle where none of the angles are right angles
• Identifying an object that represents an oblique triangle


  1. Identify the object

  2. Draw the object

  3. Identify the parts needed to apply the Sine Law

  4. Measure and label those parts

  5. Solve for the missing sides using the Laws of Sines

  6. Verify the answer using a ruler
• What I Have Learned
• What I Can Do
• Solving an oblique triangle completely

  1. Step 1

  2. Step 2
• Included angle


  The angle between two sides of a triangle
• LESSON 2: LAWS OF COSINES
• Completing statements using a given figure


  1. Identify the included angle of side a and b

  2. Identify the included angle of side b and c

  3. Identify the included angle of sides c and a
• Additional Activities
• Preparing materials

  1. Bond paper or manila paper

  2. Pencil or ballpen

  3. Protractor and ruler
• Question: What will happen if the mango tree is hit by a super typhoon?
• Instructions


  1. Draw an image of mango tree whose trunk was cut after a super typhoon hit it and fell on the ground

  2. Draw a triangle formed by the mango tree and measure the angle between the broken trunk of the tree using protractor

  3. Measure the two sides (in inch.) that include the angle

  4. Label the vertices of the triangle A, B, and C

  5. Solve for the unknown side of the tree
• From the above activity, it shows the relation of the three sides of a triangle to the cosine of an angle. This activity shows Laws of Cosines.
• Laws of Cosines


  Relate the three sides of a triangle to the cosine of the angle
• Solving for an unknown side using Laws of Cosines


  1. When side a is unknown and the other two sides b and c and their included angle A is given

  2. When side b is missing

  3. To find side c
• Solving a triangle using Laws of Cosines

  1. Given two sides and their included angle (SAS)

  2. Given three sides (SSS)
• Completing a table using a given triangle

  Round the answers to the nearest whole number
• Illustrating a triangle based on given measures and solving the remaining parts

  1. A = 50°, B = 35°

  2. A = 30°, a = 75, b = 5

  3. a = 5, b = 60, c = 8
• Statements to complete

  • The Laws of Cosines state that...
  • The Laws of Cosines can be applied when...
  • In ∆ABC, a^2 = b^2 + c^2 - ..., b^2 = ... + c^2 - 2ac cosB, c^2 = a^2 + b^2 - 2ab cosC
• Laws of Cosines are useful in finding the distance between two places
• A

  50°
• B


  35°
• A

  30°
• a

  75
• b

  5
• a

  5
• b


  60
• c

  8