Comprehensive Trigonometric Analysis and Advanced Concepts

Comprehensive Trigonometric Analysis and Advanced Concepts π 6 1. Derivation of tan
: 1.1. Trigonometric Definition of Tangent: Tangent is defined in terms of sine and cosine: sin(θ) opposite side = . cos(θ) adjacent side tan(θ) = For a 30◦ − 60◦ − 90◦ triangle, the sides are in the ratio: Opposite : Adjacent : Hypotenuse = 1 : 1.2. Calculation of tan For θ = π 6 π 6
√ 3 : 2. : (30° in radians): tan π 6 = opposite 1 =√ . adjacent 3 Rationalizing the denominator: tan π 6 √ = 3 . 3 1.3. Using the Unit Circle: On the unit circle, where the hypotenuse is 1, the coordinates of the point at ! √  π  π  3 1 , . cos , sin = 6 6 2 2 Thus: tan π 6 sin = cos π 6
π 6
= 1 √2 3 2 π 6 are: √ 1 3 =√ = . 3 3 — 2. Proof of cos(θ) = sin(90◦ − θ): 2.1. Complementary Angle Relationships: The complementary angle property in a right triangle states: θ + (90◦ − θ) = 90◦ . From this, we derive: sin(90◦ − θ) = cos(θ), cos(90◦ − θ) = sin(θ). 2.2. Geometric Proof: In a right triangle: - For ∠θ, the adjacent side is the same as the opposite side for (90◦ − θ). - The hypotenuse remains constant for both angles. Thus: opposite side to (90◦ − θ) adjacent side to θ = = sin(90◦ − θ). cos(θ) = hypotenuse hypotenuse 2.3. Unit Circle Perspective: On the unit circle: - cos(θ) corresponds to the x-coordinate of the point at angle θ. - sin(θ) corresponds to the y-coordinate. Rotating θ by 90◦ swaps the x- and y-coordinates, so: cos(θ) = sin(90◦ − θ). — 1 3. Advanced Complementary Relationships: The complementary angle property extends to all six trigonometric functions: sin(90◦ − x) = cos(x), cos(90◦ − x) = sin(x), tan(90◦ − x) = cot(x), cot(90◦ − x) = tan(x), sec(90◦ − x) = csc(x), csc(90◦ − x) = sec(x). — 4. Relationship Between sin(∠K) and cos(∠L): 4.1. Definitions in a Triangle: In a right triangle: sin(∠K) = opposite , hypotenuse cos(∠L) = adjacent . hypotenuse Since ∠K + ∠L = 90◦ , the complementary angle property implies: sin(∠K) = cos(∠L). 4.2. Generalization for Complementary Angles: The relationship holds for all complementary angles: sin(x) = cos(90◦ − x), cos(x) = sin(90◦ − x). — 5. Deriving tan(θ) Using Sine and Cosine: Using the definitions of sine and cosine: sin(θ) = opposite , hypotenuse cos(θ) = adjacent . hypotenuse The tangent function is derived as: tan(θ) = For θ = π 6: tan π 6 = 1 √2 3 2 sin(θ) . cos(θ) √ 1 3 =√ = . 3 3 — 6. Generalization of Trigonometric Identities: 6.1. Pythagorean Identities: sin2 (θ) + cos2 (θ) = 1, 1 + tan2 (θ) = sec2 (θ), 1 + cot2 (θ) = csc2 (θ). 6.2. Co-Function Identities: sin(90◦ − θ) = cos(θ), cos(90◦ − θ) = sin(θ), tan(90◦ − θ) = cot(θ), cot(90◦ − θ) = tan(θ). 2 6.3. Even-Odd Identities: sin(−θ) = − sin(θ), cos(−θ) = cos(θ), tan(−θ) = − tan(θ), csc(−θ) = − csc(θ). — 7. Applications to Real-World Problems: 7.1. Right Triangle Problems: Trigonometric functions are used to calculate unknown sides or angles: sin(θ) = opposite , hypotenuse cos(θ) = adjacent , hypotenuse 7.2. Wave Behavior: Trigonometric functions model wave-like phenomena, such as: y = A sin(ωt + ϕ), where A is amplitude, ω is angular frequency, and ϕ is phase. — 3 tan(θ) = opposite . adjacent