Problem 81 - Precalculus: Functions and Graphs

Question

Find the exact values of the six trigonometric functions of each angle, whenever possible. (a) \(90^{\circ}\) (6) \(0^{\circ}\) (c) \(7 \pi / 2\) (d) \(3 \pi\)

Step-by-Step Solution

Answer

(a) \( \sin=1, \cos=0, \tan=\text{undef}, \csc=1, \sec=\text{undef}, \cot=0 \). (b) \( \sin=0, \cos=1, \tan=0, \csc=\text{undef}, \sec=1, \cot=\text{undef} \). (c) \( \sin=-1, \cos=0, \tan=\text{undef}, \csc=-1, \sec=\text{undef}, \cot=0 \). (d) \( \sin=0, \cos=-1, \tan=0, \csc=\text{undef}, \sec=-1, \cot=\text{undef} \).

Step 1: Understanding the Unit Circle

The unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane. Trigonometric functions are defined based on the coordinates of points on the unit circle corresponding to given angles.

Step 2: Evaluate at 90 Degrees

For an angle of \(90^{\circ}\) on the unit circle, the point is \((0, 1)\). Thus, \(\sin(90^{\circ}) = 1\), \(\cos(90^{\circ}) = 0\), \(\tan(90^{\circ})\) is undefined, \(\csc(90^{\circ}) = 1\), \(\sec(90^{\circ})\) is undefined, and \(\cot(90^{\circ}) = 0\).

Step 3: Evaluate at 0 Degrees

For an angle of \(0^{\circ}\), the point is \((1, 0)\) on the unit circle. Thus, \(\sin(0^{\circ}) = 0\), \(\cos(0^{\circ}) = 1\), \(\tan(0^{\circ}) = 0\), \(\csc(0^{\circ})\) is undefined, \(\sec(0^{\circ}) = 1\), and \(\cot(0^{\circ})\) is undefined.

Step 4: Convert 7 Pi over 2 to Degrees and Evaluate

The angle \(\frac{7\pi}{2}\) in radians is equivalent to \(630^{\circ}\). This is a full circle \(360^{\circ}\) plus \(270^{\circ}\), landing on \((0, -1)\) on the unit circle. Thus, \(\sin(\frac{7\pi}{2}) = -1\), \(\cos(\frac{7\pi}{2}) = 0\), \(\tan(\frac{7\pi}{2})\) is undefined, \(\csc(\frac{7\pi}{2}) = -1\), \(\sec(\frac{7\pi}{2})\) is undefined, and \(\cot(\frac{7\pi}{2}) = 0\).

Step 5: Convert 3 Pi to Degrees and Evaluate

The angle \(3\pi\) radians is equivalent to \(540^{\circ}\). This means a full circle \(360^{\circ}\) plus \(180^{\circ}\), landing on \((-1, 0)\) on the unit circle. Thus, \(\sin(3\pi) = 0\), \(\cos(3\pi) = -1\), \(\tan(3\pi) = 0\), \(\csc(3\pi)\) is undefined, \(\sec(3\pi) = -1\), and \(\cot(3\pi)\) is undefined.

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are essential in various fields such as physics, engineering, and mathematics. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Each of these functions is based on the unit circle, which is a circle with a radius of one.

• Sine corresponds to the y-coordinate of a point on the unit circle.
• Cosine relates to the x-coordinate.
• Tangent is the ratio of sine to cosine (\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)).

The reciprocal functions are cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). They are defined as:

• Cosecant is the reciprocal of sine: \[\csc(\theta) = \frac{1}{\sin(\theta)}\]
• Secant is the reciprocal of cosine: \[\sec(\theta) = \frac{1}{\cos(\theta)}\]
• Cotangent is the reciprocal of tangent: \[\cot(\theta) = \frac{1}{\tan(\theta)}\]

These functions help describe circular and oscillatory phenomena. Understanding them is crucial for solving problems involving periodic processes.

Degrees to Radians Conversion

Angles can be measured in degrees or radians, and sometimes it's necessary to convert between the two. A complete circle is \(360^{\circ}\) degrees or \(2\pi\) radians.

• To convert degrees to radians, multiply the degree measurement by \(\frac{\pi}{180}\).
• Conversely, to convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\).

Using this conversion:- \(90^{\circ} = \frac{\pi}{2} \text{ radians}\).- \(0^{\circ} = 0 \text{ radians}\).- \(630^{\circ} = \frac{7\pi}{2} \text{ radians}\), and- \(540^{\circ} = 3\pi \text{ radians}\).
Understanding these conversions is essential for working with trigonometric functions, as they often require one unit over the other for calculation purposes.

Exact Values of Trigonometric Functions

The unit circle plays a significant role in determining the exact values of trigonometric functions. Certain angles have well-known coordinates on the unit circle, allowing us to determine their trigonometric values immediately. These are known as the exact values.

• At \(90^{\circ}\), the coordinates are \((0, 1)\). - \(\sin(90^{\circ}) = 1\) - \(\cos(90^{\circ}) = 0\) - \(\tan(90^{\circ})\) is undefined because division by zero is not possible.
• At \(0^{\circ}\), the coordinates are \((1, 0)\). - \(\sin(0^{\circ}) = 0\) - \(\cos(0^{\circ}) = 1\) - \(\tan(0^{\circ}) = 0\)
• At \(\frac{7\pi}{2}\) (or \(270^{\circ}\)), the coordinates are \((0, -1)\). - \(\sin(\frac{7\pi}{2}) = -1\) - \(\cos(\frac{7\pi}{2}) = 0\) - \(\tan(\frac{7\pi}{2})\) is undefined.
• At \(3\pi\) (or \(180^{\circ}\)), the coordinates are \((-1, 0)\). - \(\sin(3\pi) = 0\) - \(\cos(3\pi) = -1\) - \(\tan(3\pi) = 0\)

These values are derived directly from the definition of the unit circle and are pivotal in solving trigonometric problems efficiently.

Undefined Trigonometric Values

In trigonometry, there are cases where certain functions are not defined due to mathematical properties. Such cases usually arise in division by zero, which is undefined in mathematics.

• For instance, \(\tan(90^{\circ})\) is undefined because it represents \(\frac{1}{0}\).
• Similarly, \(\sec(90^{\circ})\) and \(\csc(0^{\circ})\) are undefined since these require a division by zero as well.

These undefined values mean the function cannot give a numerical result for those specific angles on the unit circle. Recognizing these cases helps prevent miscalculations when analyzing angles, especially in complex trigonometric equations.