Ch 08 - Introduction to Trigonometry10th Mathematics
Mathematical Definition and Purpose of Trigonometry
What is the literal meaning of the word 'Trigonometry' and what is its core mathematical purpose?
What is the literal meaning of the word 'Trigonometry' and what is its core mathematical purpose?
- Trigonometry is the study of relationships between the sides and angles of a triangle.
- Derived from three Greek words: 'tri' (meaning three), 'gon' (meaning sides), and 'metron' (meaning measure).
- Primary Mathematical Application: Calculating unknown distances and heights (e.g., width of a river, altitude of a balloon) without physical measurement, using side-angle relationships.
Prism Memory Hook
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Sine Ratio (sin)
Sine Ratio (sin)
State the trigonometric formula for the Sine ratio of an acute angle in a right-angled triangle.
Derivation
Geometrically defined directly from the properties of a right-angled triangle.
Component Anatomy:
- Angle A: The acute angle of reference.
- Opposite Side: The side directly facing Angle A.
- Hypotenuse: The longest side of the right-angled triangle.
Boundary Conditions / Domain:
- .
- strictly, since the hypotenuse is always the longest side of a right triangle.
Prism Memory Hook
SOH: Sine is Opposite over Hypotenuse
Cosine Ratio (cos)
Cosine Ratio (cos)
State the trigonometric formula for the Cosine ratio of an acute angle in a right-angled triangle.
Derivation
Geometrically defined directly from the properties of a right-angled triangle.
Component Anatomy:
- Angle A: The acute angle of reference.
- Adjacent Side: The side forming the angle A alongside the hypotenuse.
- Hypotenuse: The longest side of the right-angled triangle.
Boundary Conditions / Domain:
- .
- strictly, since the adjacent side cannot be longer than the hypotenuse.
Prism Memory Hook
CAH: Cosine is Adjacent over Hypotenuse
Also in this chapter
44 more cards to study in Prism
- 4Tangent Ratio (tan)
- 5Cosecant Ratio (cosec)
- 6Secant Ratio (sec)
- 7Cotangent Ratio (cot)
- 8Quotient Identities for Tangent and Cotangent
- 9Reciprocal Trigonometric Identities
- 10Notation Trap: Meaning of 'sin A'
- 11Power Notation vs Inverse Notation
- 12Independence of Trigonometric Ratios from Triangle Size
- 13Bounded Range of Sine and Cosine Functions
- 14Bounded Range of Secant and Cosecant Functions
- 15Undefined Trigonometric Ratios at 0°
- 16Undefined Trigonometric Ratios at 90°
- 17Monotonicity of Sine and Cosine in the First Quadrant
- 18Monotonicity of Tangent and Cotangent in the First Quadrant
- 19Derivation of Trigonometric Ratios for 45°
- 20Derivation of Trigonometric Ratios for 30° and 60°
- 21Complementary Angles Theorem: Sine and Cosine
- 22Complementary Angles Theorem: Tangent and Cotangent
- 23First Pythagorean Trigonometric Identity (Sine & Cosine)
- 24Second & Third Pythagorean Trigonometric Identities
- 25Algorithm: Finding All Trig Ratios from One Given Ratio
- 26Line of Sight Geometric Modeling
- 27Angle of Elevation vs. Angle of Depression
- 28SOP: Solving Basic Height and Distance Problems
- 29SOP: Double Triangle Systems - Geometric Decomposition
- 30SOP: Double Triangle Systems - Geometric Decomposition
- 31SOP: Double Triangle Systems - Algebraic Substitution
- 32Identity Proof SOP: Primary Ratio Conversion
- 33Identity Proof SOP: Conjugate Multiplication
- 34Identity Proof SOP: The '1' Substitution Trick - Setup
- 35Identity Proof SOP: The '1' Substitution Trick - Setup
- 36Identity Proof SOP: The '1' Substitution Trick - Factor & Cancel
- 37Equation Solving Trap: Extraneous Roots from Squaring
- 38Complementary Angles of Elevation: System Setup
- 39Complementary Angles of Elevation: System Setup
- 40Complementary Angles of Elevation: Algebraic Evaluation
- 41Conjugate Reciprocal Property of Sec/Tan and Cosec/Cot
- 42Parameterized Secant and Tangent Equations
- 43Telescoping Product of Complementary Tangents
- 44Algebraic Substitution Trap: Sine-Cosine Conversion
- 45Algebraic Substitution Trap: Power Shift Evaluation
- 46Fourth-Power Trigonometric Identity Reduction
- 47Sixth-Power Trigonometric Identity Reduction
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