Problem 12 - Precalculus: Functions and Graphs

Question

Let \(P\) be the point on the unit circle \(U\) that corresponds to \(t .\) Find the coordinates of \(P\) and the exact values of the trigonometric functions of \(t,\) whenever possible. (a) \(5 \pi / 2\) \((0)-\pi / 2\)

Step-by-Step Solution

Answer

(a) The coordinates for \(\frac{5\pi}{2}\) are (0, 1); trigonometric values: \(\sin=1\), \(\cos=0\). (b) The coordinates for \(-\frac{\pi}{2}\) are (0, -1); trigonometric values: \(\sin=-1\), \(\cos=0\).

Step 1: Identify the Angle

First, we need to understand the angle we are working with. We are given two angles: \(\frac{5\pi}{2}\) and \(-\frac{\pi}{2}\). Both need to be evaluated on the unit circle.

Step 2: Find Equivalent Angle for \(\frac{5\pi}{2}\)

An angle of \(\frac{5\pi}{2}\) can be simplified by subtracting multiples of \(2\pi\) (full circle) to bring it within the range of one circle \([0, 2\pi]\). \(\frac{5\pi}{2}\) is equivalent to \(\frac{5\pi}{2} - 2\pi = \frac{5\pi}{2} - \frac{4\pi}{2} = \frac{\pi}{2}\).

Step 3: Determine the Coordinates for \(\frac{\pi}{2}\)

The angle \(\frac{\pi}{2}\) corresponds to a point on the unit circle that lies directly on the positive y-axis. The coordinates for this point are \((0, 1)\).

Step 4: Calculate Trigonometric Values for \(\frac{\pi}{2}\)

For \(t = \frac{\pi}{2}\), we have the following trigonometric values: \(\sin(t) = 1\), \(\cos(t) = 0\), \(\tan(t) = \frac{1}{0}\) which is undefined. Other trigonometric functions such as \(\csc(t), \sec(t), \cot(t)\) can be derived as \(\csc(t) = 1\), \(\sec(t) = \text{undefined}\), \(\cot(t) = 0\).

Step 5: Find Equivalent Angle for \(-\frac{\pi}{2}\)

For \(-\frac{\pi}{2}\), we again consider one full rotation to adjust the angle into the standard unit circle range. This becomes \(-\frac{\pi}{2} + 2\pi = \frac{3\pi}{2}\).

Step 6: Determine the Coordinates for \(\frac{3\pi}{2}\)

The angle \(\frac{3\pi}{2}\) corresponds to a point on the unit circle that lies directly on the negative y-axis. The coordinates for this point are \((0, -1)\).

Step 7: Calculate Trigonometric Values for \(\frac{3\pi}{2}\)

For \(t = \frac{3\pi}{2}\), we have the following trigonometric values: \(\sin(t) = -1\), \(\cos(t) = 0\), \(\tan(t) = \frac{-1}{0}\) which is undefined. Other trigonometric functions such as \(\csc(t), \sec(t), \cot(t)\) are \(\csc(t) = -1\), \(\sec(t) = \text{undefined}\), \(\cot(t) = 0\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, particularly in studying the properties of triangles and the periodic nature of waves. These functions, namely sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. Trigonometric functions are expressed as:

• Sin, \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \)
• Cosine, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• Tangent, \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

Understanding these functions helps in navigating various fields of science and engineering. Moreover, other associated functions, like cosecant, secant, and cotangent, provide additional relationships and insights into the angles and distances between points.

Coordinates

Coordinates on the unit circle are crucial for understanding trigonometric functions. The unit circle is a circle with a radius of one, centered at the origin of a Cartesian coordinate system (0,0). Each point on the circle corresponds to a specific angle measured from the positive x-axis. For example:

• At angle \( \frac{\pi}{2} \), the coordinates are (0,1), positioned on the positive y-axis.
• At angle \( \pi \), the coordinates are (-1,0), located on the negative x-axis.

When you rotate around the circle, the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine. This relationship allows you to visualize how trigonometric functions behave as the angle changes.

Angle Conversion

Converting angles is an essential skill in trigonometry. Angles are often provided in radians or degrees, and understanding the relation between different measures is vital.
Radians measure angles based on the radius of the circle. Here’s how conversion works:

• To convert from degrees to radians, use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \)
• Conversely, from radians to degrees: \( \text{degrees} = \text{radians} \times \frac{180}{\pi} \)

In problems like the unit circle exercise, you may need to simplify angles, like converting \( \frac{5\pi}{2} \) into \( \frac{\pi}{2} \) by considering the circle’s full rotation (\( 2\pi \)). This makes computation straightforward within one cycle of \([0, 2\pi]\).

Sin, Cos, Tan Values

The values of sine, cosine, and tangent functions depend on the angle’s position on the unit circle. These values are crucial in various calculations and provide insight into wave patterns and other phenomena.
- **Sine** denotes the y-coordinate of a point on the unit circle: - At \( \frac{\pi}{2} \), \( \sin = 1 \) - At \( \frac{3\pi}{2} \), \( \sin = -1 \)- **Cosine** corresponds to the x-coordinate: - For both \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), \( \cos = 0 \)- **Tangent** is the ratio of sine to cosine: - At \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \), \( \tan \) is undefined due to division by zero.
Other trigonometric functions, like cosecant and secant, involve reciprocals, providing further insight into angles where sine or cosine equals zero.