MODULE 1

• DISCIPLINE • GOOD TASTE • EXCELLENCE
• Writers: Ma.Teresa C. San Andres, Veronica D. Cruz
• Cover Illustrator: Joel J. Estudillo
• Department of Education, National Capital Region, SCHOOLS DIVISION OFFICE, MARIKINA CITY
• MATHEMATICS, Quarter 4 – Module 1, The Six Trigonometric Ratios
• You can say that you have understood the lesson in this module if you can already:

  1. identify the opposite side, adjacent side, and the hypotenuse of the given right triangle<|>2. determine the formula or equation for the six trigonometric ratios<|>3. illustrate the six trigonometric ratios
• Trigonometry


  A branch of Mathematics that deals with the study of triangles and relationships between their sides and the angles between these sides
• Trigonometric ratios


  Ratios of two sides of a right triangle related to an angle
• The first three trigonometric ratios

  • Sine
  • Cosine
  • Tangent
• Opposite side


  The leg opposite to the reference angle θ
• Adjacent side


  The leg next to the reference angle θ
• Hypotenuse


  The longest side of a right triangle, opposite the right angle
• To solve a triangle

  1. Find the length of all the sides

  2. Find the measure of all the angles
• SOH-CAH-TOA


  Mnemonic to remember the formulas for sine, cosine, and tangent
• Sine (sin θ)


  Opposite side / Hypotenuse
• Cosine (cos θ)


  Adjacent side / Hypotenuse
• Tangent (tan θ)

  Opposite side / Adjacent side
• Cases for solving right triangles

  1. Given the measure of the hypotenuse and one acute angle

  2. Given the measure of hypotenuse and one of the legs

  3. Given the measure of one of the legs and one of the acute angles

  4. Given the measure of the two legs
• Case 1: Solving right triangle given the measure of the hypotenuse and one acute angle


  • Given: c = 18, m∠A = 46°

  Find: a, b, m∠B
• m∠A + m∠B = 90°


  Since ∠A and ∠B are complementary angles
• m∠B = 90° - 46° = 44°
• To find b

  b is the adjacent side of ∠A and c is the hypotenuse
• AH


  Cosine, Adjacent side, Hypotenuse
• TOA

  Tangent, Opposite side, Adjacent side
• Finding the value of missing parts in right triangles

  1. Solving right triangle given the measure of the hypotenuse and one acute angle

  2. Solving right triangle given the measure of hypotenuse and one of the legs

  3. Solving right triangle given the measure of one of the legs and one of the acute angles

  4. Solving right triangle given the measure of the two legs
• Solving right triangle given the measure of the hypotenuse and one acute angle


  1. Find 𝑚 ∠𝐵

  2. Find b using cosine ratio

  3. Find a using sine ratio
• Using calculator to find trigonometric ratios
• Cases


  • Finding the ratio given the angle
  • Finding the angle given the value of the ratio
  • Converting angles with decimals into degrees/minutes/seconds
• Solving right triangle given the measure of hypotenuse and one of the legs


  1. Find 𝑚 ∠𝐴 using sine ratio

  2. Find 𝑚 ∠𝐵 using complementary angles

  3. Find b using Pythagorean Theorem
• Solving right triangle given the measure of one of the legs and one of the acute angles

  1. Find 𝑚 ∠𝐴 using complementary angles

  2. Find c using cosine ratio

  3. Find b using Pythagorean Theorem
• Solving right triangle given the measure of the two legs

  1. Find c using Pythagorean Theorem

  2. Find 𝑚 ∠𝐵 using tangent ratio

  3. Find 𝑚 ∠𝐴 using complementary angles
• Parts of a right triangle

  • Opposite side
  • Adjacent side
  • Hypotenuse
• Trigonometric ratios


  • Sin 𝜃
  • Cos 𝜃
  • Tan 𝜃
• Sketch Making: Illustrate the six trigonometric ratios on right triangle formed by objects inside your house
• =

  Equals
• 𝑚 ∠𝐴

  Measure of angle A
• 2ndF


  Second function
• tan

  Tangent
• 1.2


  1.2
• SHIFT


  Shift