Trigonometry for physics (right triangles, components) Knowledge-Based Study

Study notes

1. Trigonometric Functions in a Right Triangle

   In physics, physical quantities having both magnitude and direction, such as displacement, velocity, and force, are represented as vectors. Analyzing vectors frequently requires working with right-angled triangles.

   Consider a right triangle containing an acute angle $\theta$. The three sides relative to $\theta$ are designated as follows:

   • Hypotenuse: The side opposite the 90° right angle; it is always the longest side.
   • Opposite side: The side situated directly across from $\theta$.
   • Adjacent side: The side located next to $\theta$ that forms one arm of the angle alongside the hypotenuse.

   The fundamental trigonometric ratios are defined as:

   $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$

   $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$

   $$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$$

   These ratios are linked by the Pythagorean theorem:

   $$(\text{adjacent})^2 + (\text{opposite})^2 = (\text{hypotenuse})^2$$

   Decomposing a Vector

   A vector $\vec{A}$ acting in a two-dimensional plane can be broken down into two mutually perpendicular parts: the horizontal component $A_x$ along the $x$-axis and the vertical component $A_y$ along the $y$-axis.

   When the angle $\theta$ is measured with respect to the horizontal ($x$-axis):

   • The adjacent side corresponds to the horizontal component: $$A_x = A \cos\theta$$
   • The opposite side corresponds to the vertical component: $$A_y = A \sin\theta$$

   Crucial Rule: Do not automatically equate $x$ with cosine and $y$ with sine. If the angle $\phi$ is measured relative to the vertical ($y$-axis):

   • $A_y = A \cos\phi$ (since the $y$-component is now adjacent to $\phi$)
   • $A_x = A \sin\phi$ (since the $x$-component is now opposite to $\phi$)

   Always verify which component lies adjacent to the reference angle and which lies opposite to it.

   Finding Magnitude and Direction

   When the perpendicular components $A_x$ and $A_y$ of a vector are known, the overall vector $\vec{A}$ can be completely reconstructed.

   1. Calculating Magnitude

   Using the Pythagorean theorem, the magnitude $A$ is calculated as:

   $$A = \sqrt{A_x^2 + A_y^2}$$

   2. Determining Direction

   The orientation angle $\theta$ relative to the positive $x$-axis is obtained using the inverse tangent function:

   $$\theta = \arctan\left(\left|\frac{A_y}{A_x}\right|\right)$$

   The signs of $A_x$ and $A_y$ determine the quadrant in which the vector lies:

   • First quadrant: $A_x > 0$, $A_y > 0$ $\rightarrow$ $\theta$ above the positive $x$-axis.
   • Second quadrant: $A_x < 0$, $A_y > 0$ $\rightarrow$ $180^\circ - \theta$.
   • Third quadrant: $A_x < 0$, $A_y < 0$ $\rightarrow$ $180^\circ + \theta$.
   • Fourth quadrant: $A_x > 0$, $A_y < 0$ $\rightarrow$ $360^\circ - \theta$ (or $-\theta$).