MODULE 2

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writers: Michelle M. Villanueva, Jocelyn O. Hulip
• Cover Illustrator: Joel J. Estudillo
• Department of Education, National Capital Region, SCHOOLS DIVISION OFFICE, MARIKINA CITY
• MATHEMATICS, Quarter 4 – Module 2, Trigonometric Ratios of Special Angles
• Trigonometric ratios of special angles


  Ratios that can be determined exactly without the use of a calculator
• sin 45°


  The exact value is √2/2
• cos 60°

  The exact value is 1/2
• tan 60°


  The exact value is √3
• Solving for the value of x


  Using trigonometric ratios and Pythagorean Theorem
• tan 30°


  The exact value is √3/3
• The simplified value of 3√2sin45° is 3/2
• Solving for the value of tan(3x-y) when x=30° and y=45°

  Using trigonometric ratios
• Solving for the value of z


  Using trigonometric ratios and Pythagorean Theorem
• Solving for the distance of the top of the ladder from the floor


  Using trigonometric ratios and given information
• Solving for the acute angle θ formed between the garden and the end of the walkway

  Using trigonometric ratios and given information
• Pythagorean Theorem


  a^2 + b^2 = c^2, where a, b, and c are the sides of a right triangle
• Trigonometric ratios of an acute angle θ

  sin θ = opposite/hypotenuse

  cos θ = adjacent/hypotenuse

  tan θ = opposite/adjacent

  csc θ = hypotenuse/opposite

  sec θ = hypotenuse/adjacent

  cot θ = adjacent/opposite
• Solving for the distance of the golf ball to the pit

  Using trigonometric ratios and given information
• Discovering the relationship between the sides of a 45°-45°-90° triangle

  Using the Pythagorean Theorem and measurements
• Discovering the relationship between the sides of a 30°-60°-90° triangle

  Using the Pythagorean Theorem and measurements
• 45°-45°-90° triangle


  A special type of isosceles right triangle where the sides are in the ratio 1:1:√2
• 30°-60°-90° triangle


  A special type of triangle where the sides are in the ratio 1:√3:2
• Special type of isosceles triangle

  • Right triangle where the interior angles are 45°, 45°, and 90° and the sides of which are always in the ratio 1:1:√2
• Using the Pythagorean relation


  1. AD^2 = AB^2 - BD^2

  2. AD = √(AB^2 - BD^2)
• The resulting triangle is a special triangle 30°-60°-90°
• The three significant angles are 30°, 45° and 60°. These angles are referred to as the special angles which can be illustrated as special right triangles.
• sin 45°

  Opposite side / Hypotenuse = √2/2
• cos 45°


  Adjacent side / Hypotenuse = √2/2
• tan 45°


  Opposite side / Adjacent side = 1
• csc 30°

  Hypotenuse / Opposite side = 2
• sec 30°

  Hypotenuse / Adjacent side = 2/√3
• cot 30°


  Adjacent side / Opposite side = √3
• cos 60°

  Adjacent side / Hypotenuse = 1/2
• csc 60°

  Hypotenuse / Opposite side = 2/√3
• cot 60°


  Adjacent side / Opposite side = 1/√3
• 45° + 45° = 1
• Using the formula sin θ = opposite side / hypotenuse


  1. sin 60° = y / 6m

  2. y = 6m * 2√3/3

  3. y = 4√3 m
• Using the formula tan θ = opposite side / adjacent side


  1. tan 60° = 75 ft / x

  2. x = 75 ft / √3

  3. x = 25√3 ft
• cos 30° = 1/2