Finding Sides with Primary Trig Ratios

Introduction

• Topic: Finding side lengths with trigonometry
• Goal: Use three primary trig ratios to solve for side lengths in right triangles.
• Review: This is a concept from last year.

Three Primary Trig Ratios

• Sine (sin)
• Cosine (cos)
• Tangent (tan)
• All trig ratios are based on a right triangle.
• Triangles with the same angles are similar, meaning side lengths are proportional.

Defining Sides Relative to an Angle

• Hypotenuse (H): Longest side, opposite the right angle.
• Opposite (O): Side opposite to the angle in question.
• Adjacent (A): Side next to the angle in question, not the hypotenuse.
• Always label sides based on the angle you are referencing.

Primary Trig Ratios Formulas

• Sine (sin): Opposite / Hypotenuse
• Cosine (cos): Adjacent / Hypotenuse
• Tangent (tan): Opposite / Adjacent
• Remember mnemonic: SOH-CAH-TOA
  • SOH: Sine = Opposite/Hypotenuse
  • CAH: Cosine = Adjacent/Hypotenuse
  • TOA: Tangent = Opposite/Adjacent

Solving for Missing Side Lengths

• Step 1: Identify and circle the angle with a known value.
• Step 2: Label sides (H, O, A) based on the circled angle.
• Step 3: Use the appropriate trig ratio (SOH-CAH-TOA).
• Step 4: Set up an equation using the trig ratio.
• Step 5: Solve using cross-multiplication or isolation.
• Example 1: Solving for side lengths with given angle and one side length.
  • Use tangent for sides O and A: tan(52°) = O/A = 12/a
  • Calculate tan(52°) and solve for a.
  • Use sine for sides O and H: sin(52°) = O/H = 12/b
  • Solve for b.
  • Ensure calculator is in degree mode.

Example Problem

• Example 2: Given angle and hypotenuse, find another side.
  • Use sine for sides O and H: sin(30°) = B/24
  • Calculate sin(30°) and solve for B.
  • Simpler calculation due to B already in numerator.
  • Result: B = 12

Conclusion

• Practice solving similar problems using trig ratios as shown in examples.
• Homework: Solve problems from textbook for further practice.