Trigonometry Math Help

Learning Objectives

• Recognize right triangles as having:
  • 2 sides and a hypotenuse
  • 2 acute angles and a right angle
• Identify a specific reference angle and its related opposite, adjacent sides, and hypotenuse
• Choose the correct trigonometric ratio based on given information
• Distinguish between sine, cosine, and tangent ratios
• Comprehend calculations to solve for unknowns

Understanding the Meaning of Trigonometry

• Trigonometry: Study of measurement of three-sided figures (triangles)
• Ancient Greek mathematicians observed that right triangles with two sides in the same ratio have equal angles

Trigonometric Ratios

• Sine (sin): opposite/hypotenuse
• Cosine (cos): adjacent/hypotenuse
• Tangent (tan): opposite/adjacent

Helpful Hints for Choosing the Appropriate Ratio:

1. Select the acute angle from which the diagram is viewed.
2. Label sides as opposite, adjacent, and hypotenuse:
   • Opposite: across from the angle
   • Adjacent: beside the angle
   • Hypotenuse: longest side, opposite the right angle
3. Based on known and unknown values, select the correct ratio.

Example Problem

• Given right triangle with:

  • Known: Hypotenuse = 12 cm
  • Unknown: Adjacent side

• Use cosine ratio:

  [\text{cos} 35^\circ = \frac{\text{Adjacent}}{12}]

  Solving gives the adjacent side ≈ 9.8 cm

Using Trigonometric Ratios

• Correctly using trigonometric ratios allows finding:
  • Every possible angle from side ratios
  • Every possible side measurement from angles
• Focus is on right triangles, though trigonometry applies to non-right triangles

Explore It Mode Examples

1. Using R as reference angle:
   • sin R to find opposite side
   • cos R to find hypotenuse
2. Using S as reference angle:
   • tan S to find adjacent side
   • Known opposite and adjacent sides, find angle S using tan S