PC

PHYS1702 W1 L2

Right-Angled Triangle Fundamentals

  • A right-angled triangle has one internal angle of 90^\circ.
  • Side opposite the right angle is the hypotenuse (longest side), traditionally labelled c.
  • Remaining sides (often a and b) form the right angle.
  • Pythagoras’ Theorem connects side lengths:
    • a^2 + b^2 = c^2
    • Rearranged for the hypotenuse: c = \sqrt{a^2 + b^2}

Naming Sides Relative to an Angle \theta

  • When one of the non-right angles is specified (labelled \theta):
    • Adjacent (\textbf{Adj}) – the side that touches \theta and is not the hypotenuse.
    • Opposite (\textbf{Opp}) – the side across from \theta.
    • Hypotenuse (\textbf{Hyp}) – always opposite the right angle.
  • Orientation does not matter; relabel sides each time \theta moves.
  • Four example orientations were shown to emphasise re-labelling:
    1. \theta in the lower-left corner → left side is Adj, bottom is Opp.
    2. \theta in upper-right → right side Adj, top Opp.
    3. \theta in other corners likewise; Hyp stays the longest side.

Key Trigonometric Relationships

  • Two must-memorise equations link the non-hypotenuse sides to \theta and the hypotenuse H:
    • Adjacent: \text{Adj} = H \cos \theta
    • Opposite: \text{Opp} = H \sin \theta
  • These appear almost identical—difference is only sine vs cosine.

Memory Aid (Alphabet Trick)

  • "Adjacent" (A) and "Cos" (C) both start near the beginning of the alphabet ⇒ pair them.
  • "Opposite" (O) and "Sin" (S) reside toward the alphabet’s end ⇒ pair them.
  • Any other mnemonic is acceptable; the author urges memorisation.

Building Vector Components

  • A 2-D vector \vec v of magnitude v and direction \theta (measured from a chosen axis) can be decomposed via a right-angled triangle whose hypotenuse is \vec v.
  • Procedure:
    1. Draw a line from the vector’s tail parallel to the y-axis → creates the x component side.
    2. Draw a line parallel to the x-axis → creates the y component side.
    3. Identify which of these legs is Adjacent or Opposite to \theta.
    4. Apply \text{Adj} = v \cos \theta or \text{Opp} = v \sin \theta accordingly.
    5. Attach correct signs ( + / – ) based on the arrow’s direction relative to positive axes.
  • Component notation:
    • v_x – projection along the x-axis.
    • v_y – projection along the y-axis.
    • Reconstructing the vector: \vec v = vx \hat i + vy \hat j.
    • Magnitude check: v = \sqrt{vx^2 + vy^2} (Pythagoras revisited).

Worked Examples & Sign Convention

  1. First-quadrant vector (pointing up & right, \theta from x-axis):
    • v_x = v \cos \theta (+ sign, rightward)
    • v_y = v \sin \theta (+ sign, upward)
  2. Same arrow but \theta measured from the y-axis:
    • Here vx is Opposite → vx = v \sin \theta
    • vy is Adjacent → vy = v \cos \theta
    • Signs remain positive (still in first quadrant).
  3. Third-quadrant style (left & down):
    • Draw legs leftward and downward.
    • vx Adjacent → vx = - \, v \cos \theta (negative; left)
    • vy Opposite → vy = - \, v \sin \theta (negative; down)
  4. Second-quadrant style (left & up):
    • v_x = - \, v \sin \theta (Opposite, negative)
    • v_y = v \cos \theta (Adjacent, positive)
  • Moral: determine Adj/Opp first, then attach +/– for direction.

Practical & Conceptual Takeaways

  • Component decomposition converts a single vector problem into independent horizontal & vertical parts.
  • Sign convention encodes direction and is essential for vector addition, kinematics, forces, etc.
  • Memorising \text{Adj} = H \cos \theta and \text{Opp} = H \sin \theta expedites trigonometry in physics and engineering contexts.
  • Always draw a quick sketch; mis-identifying Adj/Opp is a common mistake.