Trigonometry/Vectors in the Plane

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In practice, one of the most useful applications of trigonometry is in calculations related to vectors, which are frequently used in Physics. A vector is a quantity which has both magnitude (such as three or eight) and direction (such as north or 30 degrees south of east). It is represented in diagrams by an arrow, often pointing from the origin to a specific point.

A plane vector \vec{A} can be expressed in two ways -- as the sum of a horizontal vector of magnitude A_x and a vertical vector of magnitude A_y, or in terms of its angle \theta and magnitude \left|\vec{A}\right| (or simply A). These two methods are called "rectangular" and "polar" respectively.

Rectangular to Polar conversionEdit

For simplicity, assume \vec{A} is in the first quadrant and has x-component A_x and y-component A_y (which will necessarily be positive). Given these components, we want to find the angle \theta and the magnitude A.

If we draw all three of these vectors, they form a right triangle. It is easy to see that \tan \theta = \frac{A_y}{A_x}, or \theta = \arctan \frac{A_y}{A_x} (A vector with an angle of zero is defined to be pointing directly to the right.) Furthermore, by the Pythagorean Theorem, A_x\,^2 + A_y\,^2=A^2, or A=\sqrt{A_x\,^2 + A_y\,^2}.

Polar to Rectangular conversionEdit

This is essentially the same problem as above, but in reverse. Here, \theta and A are known and we want to calculate the values of A_x and A_y.

Using the same triangle as above, we can see that \cos \theta = \frac{A_x}{A}, or A_x=A \cos \theta. Also, \sin \theta = \frac{A_y}{A}, or A_y=A \sin \theta.

Review of conversionsEdit

  • \theta = \arctan \frac{A_y}{A_x}
  • \left|\vec{A}\right|=A=\sqrt{A_x\,^2 + A_y\,^2}
  • A_x=A \cos \theta
  • A_y=A \sin \theta

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