Lecture on Pythagoras's Theorem and Trigonometrical Ratios

Objectives

• Describe Pythagoras's Theorem
• Describe primary trigonometrical ratios
• Find one ratio given another ratio
• Describe relationships between the ratios

Pythagoras's Theorem

• Applicable only to right-angled triangles.
• States: The square of the hypotenuse is equal to the sum of the squares of the other two sides.
• Formula: ( a^2 + b^2 = c^2 ) (where ( c ) is the hypotenuse)
• Used to calculate unknown lengths in a triangle.
• Example: 3-4-5 Triangle:
  • Squares: ( 3^2 + 4^2 = 5^2 )
  • ( 3^2 = 9, 4^2 = 16, 5^2 = 25 )
  • 9 + 16 = 25
• Pythagorean Triples:
  • Examples: (3, 4, 5), (6, 8, 10), (9, 12, 15)

Naming Triangle Sides

• Hypotenuse (H): Longest side, opposite the right angle.
• Opposite (O): Side across from the angle.
• Adjacent (A): Side next to the angle.

Trigonometric Ratios

• Sine (sin): ( \frac{\text{Opposite}}{\text{Hypotenuse}} )
• Cosine (cos): ( \frac{\text{Adjacent}}{\text{Hypotenuse}} )
• Tangent (tan): ( \frac{\text{Opposite}}{\text{Adjacent}} )
• Mnemonic: SOH-CAH-TOA

Example Problems

• Ratio Calculation:
  • Given: ( \sin \theta = \frac{3}{5} )
  • Calculate: ( \cos \theta = \frac{4}{5}, \tan \theta = \frac{3}{4} )
• Finding Hypotenuse:
  • Example Triangle: Sides 2 and 2
  • Hypotenuse: ( \sqrt{8} )

Trigonometric Relationships

• Complementary Angles:
  • If ( a + b = 90^\circ ), then ( \sin a = \cos b )
• Special Angles:
  • ( \sin 45^\circ = \cos 45^\circ = \frac{1}{\sqrt{2}} )
  • ( \tan 45^\circ = 1 )

Finding Ratios Given One

• Example:
  • Given: ( \sin \theta = \frac{10}{11} )
  • Find: ( \cos \theta = \frac{\sqrt{21}}{11}, \tan \theta = \frac{10}{\sqrt{21}} )

Applications

• Often used in multiple-choice questions to find unknown sides or angles based on given trigonometric ratios.
• Important to label sides correctly to ensure accurate calculation.

Key Takeaways

• Pythagoras's Theorem and trigonometric ratios are fundamental tools in geometry.
• Understanding these concepts helps in solving various mathematical and real-world problems related to right-angled triangles.