Problem 58 - Precalculus Mathematics for Calculus

Question

Now that you know the terminal point determined by \(t=\pi / 6,\) use symmetry to find the terminal point determined by \(t=\pi / 3\) (see the figure). Explain your reasoning. CAN'T COPY THE GRAPH

Step-by-Step Solution

Answer

The terminal point for \(t=\pi/3\) is \((1/2, \sqrt{3}/2)\).

Step 1: Determine Terminal Point at t=π/6

The terminal point determined by \(t = \pi/6\) on the unit circle corresponds to the point \((\cos(\pi/6), \sin(\pi/6))\). We know \(\cos(\pi/6) = \sqrt{3}/2\) and \(\sin(\pi/6) = 1/2\), so the terminal point is \((\sqrt{3}/2, 1/2)\).

Step 2: Understand Symmetry on the Unit Circle

The unit circle is symmetrical across different axes. For example, reflection over the line \(y = x\) or changes in the order of sine and cosine values can help find new points.

Step 3: Apply Symmetry to Find Terminal Point at t=π/3

To find the terminal point for \(t = \pi/3\), we can use symmetry. Notice that \(\pi/3\) is the complement of \(\pi/6\) in the first quadrant, where sine and cosine values can be swapped. Thus, the terminal point at \(t = \pi/3\) should be \((\cos(\pi/3), \sin(\pi/3))\) = \((1/2, \sqrt{3}/2)\).

Key Concepts

Trigonometric Functions

Trigonometric functions form the backbone of understanding angles and their relationships in mathematics, especially in the context of a unit circle. The unit circle is a circle with a radius of one centered at the origin of a coordinate system. Trigonometric functions such as sine and cosine represent coordinates on this circle. They are used to define the coordinates of points that lie on the circle's circumference depending on the angle.

• The function \(\cos(t)\) represents the x-coordinate.
• The function \(\sin(t)\) represents the y-coordinate.

In the exercise, for the angle \(t = \pi/6\), these functions yield the terminal point \((\sqrt{3}/2, 1/2)\). These values are derived from values that solve the special right triangles or unit circle characteristics. As the angle changes, these functions help describe movement around the circle, representing continuous periodic patterns.

Symmetry in Mathematics

Symmetry is a powerful concept in mathematics that simplifies the study of complex systems. The unit circle is a great example where symmetry plays a major role.

• Symmetry allows us to understand how swapping angles can provide solutions quickly.
• In the unit circle, the symmetry about lines such as \(y = x\) helps easily determine coordinates.

For example, in the problem context, the angle \(t = \pi/3\) is a complement to \(t = \pi/6\) in the first quadrant. This symmetry flips the x and y coordinates due to equal complementary angle reflections, leading us from \((\sqrt{3}/2, 1/2)\) to \((1/2, \sqrt{3}/2)\). This approach is notably beneficial when working with larger angles and in constructing transformations or solving trigonometric identities.

Angle Measurement in Radians

Understanding angle measurements in radians is crucial for working with the unit circle and trigonometric functions. Unlike degrees, radians offer a natural and mathematical way of representing angles.

• A radian measures an angle based on the radius of a circle. When the arc of a circle is equal to the circle’s radius, this angle is precisely one radian.
• Using the formula \(2\pi\) radians equals a full circle, you can determine the radian measure by relating it to the circle's circumference.

In our context, angles like \(\pi/6\) and \(\pi/3\) denote specific sections of the circle, each representing a fraction of \(\pi\). Therefore, using radians simplifies calculations and visualizes rotational movement on the circle. This is essential because it ties directly into the symmetry discussed and ensures consistency in trigonometric analysis.