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Magazine Chapter 5: Problem 26
Find the value of each of the six trigonometric functions (if it is defined) at the given real number \(t\). Use your answers to complete the table. $$t=\frac{\pi}{2}$$ (TABLE CAN'T COPY).
Short Answer
Expert verified
Sine = 1, Cosine = 0, Tangent = undefined, Cosecant = 1, Secant = undefined, Cotangent = 0.
Step by step solution
01
Understanding the angle
We start by understanding that the angle given is in radians. The angle is \( t = \frac{\pi}{2} \), which is equivalent to 90 degrees in the unit circle.
02
Sine function value
At \( t = \frac{\pi}{2} \), the sine function, \( \sin(t) \), represents the y-coordinate of the corresponding point on the unit circle. At \( 90^\circ \), this point is \((0, 1)\), so \( \sin\left(\frac{\pi}{2}\right) = 1 \).
03
Cosine function value
For the cosine function, \( \cos(t) \), it represents the x-coordinate on the unit circle. At \( 90^\circ \), the point is \((0, 1)\), so \( \cos\left(\frac{\pi}{2}\right) = 0 \).
04
Tangent function value
The tangent function is defined as \( \tan(t) = \frac{\sin(t)}{\cos(t)} \). Substituting the sine and cosine values, we get \( \tan\left(\frac{\pi}{2}\right) = \frac{1}{0} \), which is undefined because division by zero is not possible.
05
Cosecant function value
The cosecant function is the reciprocal of sine, \( \csc(t) = \frac{1}{\sin(t)} \). Since \( \sin\left(\frac{\pi}{2}\right) = 1 \), \( \csc\left(\frac{\pi}{2}\right) = 1 \).
06
Secant function value
The secant function is the reciprocal of cosine, \( \sec(t) = \frac{1}{\cos(t)} \). With \( \cos\left(\frac{\pi}{2}\right) = 0 \), \( \sec\left(\frac{\pi}{2}\right) \) is undefined because division by zero is not possible.
07
Cotangent function value
The cotangent function is the reciprocal of tangent, \( \cot(t) = \frac{1}{\tan(t)} \). Since \( \tan\left(\frac{\pi}{2}\right) \) is undefined, \( \cot\left(\frac{\pi}{2}\right) = 0 \) based on the relation with sine and cosine.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Unit Circle
The unit circle is a fundamental concept in trigonometry, crucial for understanding angles and their corresponding points. Imagine a circle with a radius of one unit, centered at the origin of a coordinate plane. This circle allows us to easily visualize trigonometric functions. Each angle, measured in radians or degrees, corresponds to a specific point on this circle. For example:
- At 0 radians (or 0 degrees), we're at point (1, 0).
- At \(\frac{\pi}{2}\) radians (or 90 degrees), we find ourselves at (0, 1).
- At \(\pi\) radians (or 180 degrees), the point is (-1, 0).
- At \(\frac{3\pi}{2}\) radians (or 270 degrees), the point lies at (0, -1).
- And finally, at \(2\pi\) radians (or 360 degrees), we return to (1, 0).
Radians
Radians offer a natural way to measure angles in mathematics and are defined by the arc length of a circle. One radian is the angle created when the radius is wrapped along the arc length of the circle. In a full circle, there are \(2\pi\) radians. Thus, this measure is rooted in the relationship between a circle's circumference and its radius. To convert between degrees and radians, use the conversion factor \(\pi\) radians equals 180 degrees. For example:
- 90 degrees converts to \(\frac{\pi}{2}\) radians.
- 180 degrees equals \(\pi\) radians.
- 360 degrees becomes \(2\pi\) radians.
Reciprocal Trigonometric Functions
Reciprocal trigonometric functions are based on the basic sine, cosine, and tangent functions, extending their utility by considering their inverses. These include:
- Cosecant (\(\csc(t)\)), which is the inverse of sine: \(\csc(t) = \frac{1}{\sin(t)}\).
- Secant (\(\sec(t)\)), the inverse of cosine: \(\sec(t) = \frac{1}{\cos(t)}\).
- Cotangent (\(\cot(t)\)), which inversely relates to tangent: \(\cot(t) = \frac{1}{\tan(t)}\).
Undefined Values in Trigonometry
In trigonometry, not all functions yield defined values for every angle. A function can become undefined primarily due to division by zero. When this occurs, it means the function lacks a finite value at that specific point.
- For example, tangent \(\tan(t) = \frac{\sin(t)}{\cos(t)}\) becomes undefined when \(\cos(t) = 0\), such as at \(\frac{\pi}{2}\) radians.
- The secant function (inverse cosine) \(\sec(t) = \frac{1}{\cos(t)}\) becomes undefined for similar reasons, like at \(\frac{\pi}{2}\) radians where cosine equals zero.
- Conversely, the cosecant \(\csc(t)\) and cotangent \(\cot(t)\) functions are undefined when \(\sin(t) = 0\), such as at 0 or \(\pi\) radians.
