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Fundamentals of Right-Angled Triangle Trigonometry

  • Right-Angled Triangle Structure:

    • A triangle containing one angle measuring exactly 90∘90^\circ (a right angle).
    • The side opposite to the right angle is termed the hypotenuse, which is always the longest side of the right-angled triangle.
    • The remaining two sides are designated as the perpendicular (opposite side) and the base (adjacent side) relative to the specific acute reference angle being evaluated.

  • Pythagoras Theorem:

    • In any right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides:
    • Hypotenuse2=Base2+Perpendicular2\text{Hypotenuse}^2 = \text{Base}^2 + \text{Perpendicular}^2

  • Trigonometric Ratio Definitions:

    • Sine (sin⁡\sin) of an acute angle is defined as the ratio of the length of the side opposite to that angle to the length of the hypotenuse:
    • sin⁡(θ)=Side opposite to θHypotenuse\sin(\theta) = \frac{\text{Side opposite to } \theta}{\text{Hypotenuse}}
    • Cosine (cos⁡\cos) of an acute angle is defined as the ratio of the length of the side adjacent to that angle to the length of the hypotenuse:
    • cos⁡(θ)=Side adjacent to θHypotenuse\cos(\theta) = \frac{\text{Side adjacent to } \theta}{\text{Hypotenuse}}

Worked Example: Evaluating Trigonometric Ratios

  • Problem Statement:

    • In ΔABC\Delta ABC, right-angled at vertex BB:
    • Given side length AB=24 cmAB = 24\,\text{cm}
    • Given side length BC=7 cmBC = 7\,\text{cm}
    • Determine the values of:
    • (i) sin⁡(A)\sin(A) and cos⁡(A)\cos(A)
    • (ii) sin⁡(C)\sin(C) and cos⁡(C)\cos(C)

  • Geometric Representation:

Right-angled triangle ABC with right angle at B, AB of 24 cm, BC of 7 cm, and hypotenuse AC of 25 cm

  • Step 1: Computation of the Unknown Hypotenuse (ACAC):

    • Identify the given triangle parameters:
    • In ΔABC\Delta ABC, ∠B=90∘\angle B = 90^\circ
    • Side adjacent/perpendicular AB=24 cmAB = 24\,\text{cm}
    • Side adjacent/perpendicular BC=7 cmBC = 7\,\text{cm}
    • Apply the Pythagoras Theorem:
    • AC2=AB2+BC2AC^2 = AB^2 + BC^2
    • Substitute the known lengths into the equation:
    • AC2=(24)2+72AC^2 = (24)^2 + 7^2
    • Evaluate the squares:
    • (24)2=576(24)^2 = 576
    • 72=497^2 = 49
    • AC2=576+49=625AC^2 = 576 + 49 = 625
    • Calculate the principal square root to find side length ACAC:
    • AC=625AC = \sqrt{625}
    • AC=25 cmAC = 25\,\text{cm}

  • Step 2: Evaluation of Ratios with Respect to Angle AA:

    • Identification of sides relative to reference angle AA:
    • Opposite side to ∠A\angle A is BC=7 cmBC = 7\,\text{cm}
    • Adjacent side to ∠A\angle A is AB=24 cmAB = 24\,\text{cm}
    • Hypotenuse is AC=25 cmAC = 25\,\text{cm}
    • Calculation of sin⁡(A)\sin(A):
    • sin⁡(A)=BCAC\sin(A) = \frac{BC}{AC}
    • sin⁡(A)=725\sin(A) = \frac{7}{25}
    • Calculation of cos⁡(A)\cos(A):
    • cos⁡(A)=ABAC\cos(A) = \frac{AB}{AC}
    • cos⁡(A)=2425\cos(A) = \frac{24}{25}

  • Step 3: Evaluation of Ratios with Respect to Angle CC:

    • Identification of sides relative to reference angle CC:
    • Opposite side to ∠C\angle C is AB=24 cmAB = 24\,\text{cm}
    • Adjacent side to ∠C\angle C is BC=7 cmBC = 7\,\text{cm}
    • Hypotenuse is AC=25 cmAC = 25\,\text{cm}
    • Calculation of sin⁡(C)\sin(C):
    • sin⁡(C)=ABAC\sin(C) = \frac{AB}{AC}
    • sin⁡(C)=2425\sin(C) = \frac{24}{25}
    • Calculation of cos⁡(C)\cos(C):
    • cos⁡(C)=BCAC\cos(C) = \frac{BC}{AC}
    • cos⁡(C)=725\cos(C) = \frac{7}{25}

Analytical Summary of Complementary Angle Relations

  • Complementary Relationship in Right Triangles:
    • Since ∠B=90∘\angle B = 90^\circ, the acute angles AA and CC are complementary:
    • $$A + C = 90^