Episode 3: Trigo-Nom-Nom-Nom-etry

Episode title: Trigo-nom-nom-nom-etry

Preview

In the last episode, DOT learned how to walk directly to their snacks using $x$-components and $y$-components of vectors. In this episode, DOT will learn how to walk in a particular direction using angles instead.

Play

[DOT game still in development!]

Practice

To give DOT directions to their snacks, you gave them instructions to walk at a particular angle. In general, it will not be possible to guess the angle of a vector. Instead, we must perform a short calculation.

Calculating the angle $\theta_V$ of a vector $\vec{V}$ has three steps:

1. We already performed the first step in the previous episode. If the $x$-component and $y$-component of our vector $\vec{V}$ are given by $\langle V_x, V_y \rangle$, then for the first step, we must determine the quotient $\dfrac{V_y}{V_x}$.
2. For the second step, we must take the inverse tangent (also called the arctangent) of the quotient from Step 1: $\tan^{-1}\left(\dfrac{V_y}{V_x} \right)$.
   • Make sure your calculator is set to degree mode (instead of radian mode) for this step.
   • On a lot of calculators, the $\tan^{-1}$ button is accessed by pressing the 2^nd button and then the $\tan$ button.
3. Our calculator answer from Step 2 will always be between -90° and 90°. Usually, we want to measure our angle starting from the positive $x$-direction for vectors pointing in any direction, so we need to convert the calculator answer from Step 2 to the standard angle.
   • If the vector points to the upper right, the standard angle is the calculator answer from Step 2.
   • If the vector points to the upper left or lower left, the standard angle is the calculator answer from Step 2 (which might be negative) plus 180°.
   • If the vector points to the lower right, the calculator answer will be negative and the standard answer is the calculator answer from Step 2 plus 360°.

For example, assume our vector $\vec{V}$ is $\langle -3, 2 \rangle$. We can use our calculators to perform Step 1 and Step 2 at once by evaluating $\tan^{-1}\left(\dfrac{2}{-3} \right)$, which is approximately −33.69°. To determine the standard angle in Step 3, we first notice that since $V_x$ is negative and $V_y$ is positive, our vector points to the upper left, which means that angle $\theta_V$ will be somewhere between 90° and 180°. Therefore, we add 180° to the answer the calculator gives us. Since 180° + (−33.69°) = 146.31°, this is our standard angle $\theta_V$.

You don’t have to already be familiar with the inverse tangent function to perform these calculations. If you have seen the tangent function in your trigonometry class, you might remember that the tangent of an angle in a right triangle is given by taking the length of the opposite side over the length of the adjacent side: $\tan(\theta) = \dfrac{O}{A}$. We can then solve for $\theta$ by applying $\tan^{-1}$ to both sides of this equation. With vectors in the plane, we have to be a little bit more careful since they can point in any direction. This is the reason for Step 3 in our calculations above.

Apply

In real-world applications, it’s not always convenient to use standard angles. Instead, we will compare the directions of vectors to the east-west direction. For example, if a vector is pointing to the upper left, we say that it is pointing north of west. In this case, due west is 180º, so we would compare the standard angle with 180º to determine how many degrees north of west the vector is pointing. Draw a picture to determine angles and directions for other vectors!