Question

Alice Through the Looking Glass by Lewis Carroll introduced strange new worlds where time ran backwards. Your challenge is to imagine a unit circle in which a positive rotation is defined to be clockwise. Assume the coordinate system remains as we know it. a) Draw a unit circle in which positive angles are measured clockwise from \((0,1) .\) Label where \(R\left(\frac{\pi}{6}\right), R\left(\frac{5 \pi}{6}\right), R\left(\frac{7 \pi}{6}\right)\) and \(\mathrm{R}\left(\frac{11 \pi}{6}\right)\) are on your new unit circle. b) What are the coordinates for the new \(\mathrm{R}\left(\frac{\pi}{6}\right)\) and \(\mathrm{R}\left(\frac{5 \pi}{6}\right) ?\) c) How do angles in this new system relate to conventional angles in standard position? d) How does your new system of angle measure relate to bearings in navigation? Explain.

Short answer

Positive angles are clockwise from (0,1). Coordinates for \(R\left(\frac{\pi}{6}\right)\) are (1/2, \sqrt{3}/2) and for \(R\left(\frac{5\pi}{6}\right)\) are (-1/2, \sqrt{3}/2). This system is analogous to navigation bearings.

Step-by-step solution

Draw a Unit Circle

Start by drawing a unit circle with the center at the origin (0,0). Ensure the radius is 1 unit.

Define Positive Rotation

In this unit circle, positive angles are measured clockwise. This is different from the standard counterclockwise direction.

Identify the New Starting Point

The new starting point for measuring angles is (0,1), which is at the top of the unit circle.

Plot R\left(\frac{\pi}{6}\right)

Since positive rotation is clockwise, \(R\left(\frac{\pi}{6}\right)\) will be at an angle of \(\frac{\pi}{6}\) radians clockwise from (0,1). This lands at coordinates (1/2, \sqrt{3}/2).

Plot R\left(\frac{5\pi}{6}\right)

Similarly, rotate \(\frac{5\pi}{6}\) radians clockwise from (0,1). This lands at coordinates (-1/2, \sqrt{3}/2).

Plot R\left(\frac{7\pi}{6}\right)

Rotate \(\frac{7\pi}{6}\) radians clockwise from (0,1). This lands at coordinates (-1/2, -\sqrt{3}/2).

Plot R\left(\frac{11\pi}{6}\right)

Rotate \(\frac{11\pi}{6}\) radians clockwise from (0,1). This lands at coordinates (1/2, -\sqrt{3}/2).

Identify Coordinates for Specific Points

\(R\left(\frac{\pi}{6}\right)\) is (1/2, \sqrt{3}/2) and \(R\left(\frac{5\pi}{6}\right)\) is (-1/2, \sqrt{3}/2).

Relate New System to Standard Position

Angles in this new system are the negative of the conventional angles measured counterclockwise. For example, \(\frac{\pi}{6}\) radians clockwise is equivalent to \(-\frac{\pi}{6}\) radians counterclockwise.

Relate to Navigation Bearings

In navigation, bearings are measured clockwise from the north. This means the new system of angle measurement is analogous to navigation bearings, where the positive direction is also clockwise.

Key concepts

unit circle

Imagine a unit circle as a circle with a radius of 1 unit in the coordinate system, centered at the origin (0,0). This circle is crucial in trigonometry for defining sine and cosine values for angles, which helps us understand angle rotations and their effects on coordinates. To draw it, place the center at the origin and use a radius of 1. All points on this circle are at an equal distance (1 unit) from the center.

angle measurement

Angle measurement in trigonometry usually uses radians or degrees. In our imagined scenario, angles are measured clockwise, unlike the standard counterclockwise direction. This means a positive angle moves in the direction of a clock's hands. Start measuring from the point (0,1), located at the top of the unit circle. For instance, an angle of \(\frac{\text{\textcolor{red}{\pi}}}{\text{\textcolor{red}{6}}}\) radians represents one-sixth of a full circle, but in the clockwise direction.

coordinate system

The coordinate system consists of the X-axis (horizontal) and Y-axis (vertical) intersecting at the origin (0,0). In trigonometry, we place angles and their measurements on this plane to determine the exact coordinates of resulting points. For example, when we rotate clockwise by \(\frac{\text{\textcolor{red}{\pi}}}{\text{\textcolor{red}{6}}}\) from (0,1), we end up at the coordinates (0.5, \sqrt{3}/2).

clockwise rotation

Clockwise rotation is the movement in the direction of a clock's hands. In our case, positive angles are defined in this direction. This changes how we interpret angle measurements. For example, an angle of \(\frac{\text{\textcolor{red}{5\text{\pi}}}}{\text{\textcolor{red}{6}}}\) radians clockwise differs from traditional trigonometry, where positive rotations are counterclockwise.

navigation bearings

Navigation bearings also use clockwise rotation for positive directions, starting from the north point (top of the coordinate system). This is similar to our imagined unit circle setup. For instance, a bearing of 30 degrees matches precisely a \(\frac{\text{\textcolor{red}{\pi}}}{\text{\textcolor{red}{6}}}\) clockwise rotation. Therefore, our imagined system relates well to how navigators plot course directions.