Video 5: Introduction to Trigonometry by MIT OpenCourseWare

Summary by www.lecturesummary.com: Video 5: Introduction to Trigonometry by MIT OpenCourseWare

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  • Trigonometry Overview

    Why Learn Trigonometry?

    Although Euclidean geometry is basic, it frequently calls for creative thinking. Geometry shouldn't be a hindrance in disciplines like physics. A more methodical approach to studying geometry is provided by trigonometry and coordinate geometry. In contrast to the "innovative leap" that geometry frequently requires, trigonometry enables reasoning about geometric problems in a more algebraic manner, which can feel more mechanical.

    What is Trigonometry?

    It is the algebraic study of angles.

    Measuring Angles

    Degrees

    A fundamental idea in Euclidean geometry that deals with circles:

    • A full circle is 360 degrees.
    • In a circle of radius r, the arc length with angle α° is (α/360) * 2πr. An 'annoying expression' for the arc length formula is involved here.

    Radians

    Introduced as an alternative unit of measurement for angles:

    • Defined so that the angle itself in radians is the expression 2π α / 360, where α is in degrees.
    • It's crucial to think in terms of radians, and it's frequently the better or more intuitive option. Instead of translating, an analogy is made to thinking directly in a new language.
    • The arc's length with angle θ is just r * θ in radians. The reason radians are more natural is demonstrated by this easier formula.
    • Examples of conversion: 90° = π/2 radians, 180° = π radians, and 360° = 2π radians.
    • Because this is the more intuitive way of thinking, it is usually assumed that θ is in radians when it is written as the angle.

    Trigonometric Ratios

    Defining Ratios based on Right-Angled Triangles

    Considering an angle θ (in radians) within a circle of radius r, drop a perpendicular to form a right-angled triangle:

    • Sides are identified as hypotenuse (the radius), adjacent (to the angle), and opposite (to the angle).
    • By using triangle similarity, it is demonstrated that the side ratio is solely a function of the angle and is independent of the triangle's size.
    • The definition of sine (sin θ) is the ratio of the opposite side to the hypotenuse.
    • The length of the opposite side is r * sin θ if the hypotenuse is r.
    • The definition of cosine (cos θ) is the ratio of the adjacent side to the hypotenuse.
    • The length of the adjacent side is r * cos θ if the hypotenuse is r.
    • These ratios (cos and sin) are characteristics of the angle and are independent of the triangle's sides.

    First Trigonometric Identity derived from Pythagoras Theorem

    Trigonometric Identities

    The Pythagorean theorem states that for any right-angled triangle:

    • opposite² + adjacent² = hypotenuse².
    • (opposite/hypotenuse)² + (adjacent/hypotenuse)² = 1 is obtained by dividing the Pythagorean theorem equation by hypotenuse².
    • sin² θ + cos² θ = 1 results from substituting the definitions of sin θ and cos θ.
    • Note on notation: sin² θ does not mean sin(θ²), but (sin θ)².