Problem 24 - Precalculus : Building Concepts and Connections

Question

Find the exact values of all the trigonometric functions for the giocn calues of \(t .\) If a certain value is undefined, state sa Do not use a calculator. $$t=6 \pi$$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions for \(t = 6\pi\) are: \(\sin(6\pi) = 0\), \(\cos(6\pi) = 1\), \(\tan(6\pi) = 0\), \(\csc(6\pi) = \)undefined, \(\sec(6\pi) = 1\), \(\cot(6\pi) = \)undefined.

Step 1: Find Exact Values of Cosine and Sine

The cosine function, \(\cos(t)\), oscillates between 1 and -1 over the interval of \(0\) to \(2\pi\) and this pattern repeats for every period of \(2\pi\). Given \(t=6\pi\), this falls exactly onto a multiple of \(2\pi\) periods. The cosine of any multiple of \(2\pi\) is always 1. Therefore, \(\cos(6\pi) = 1\).\nThe sine function, \(\sin(t)\), also oscillates over the interval of \(0\) to \(2\pi\), but it starts at 0, increases to 1 at \(\pi/2\), decreases to 0 at \(\pi\), decreases further to -1 at \(3\pi/2\), and returns to 0 at \(2\pi\). This pattern repeats for every \(2\pi\) periods. So, \(\sin(6\pi) = 0\).

Step 2: Find Exact Values for Tangent

The tangent function, \(\tan(t)\), is defined as the ratio of sine to cosine, that is, \(\tan(t) = \sin(t)/\cos(t)\). For \(t = 6\pi\), we have \(\sin(6\pi) = 0\) and \(\cos(6\pi) = 1\). Therefore, \(\tan(6\pi) = 0/1 = 0\).

Step 3: Find Exact Values for Cosecant, Secant, and Cotangent

The cosecant function, \(\csc(t)\), is the reciprocal of the sine function. That is, \(\csc(t) = 1/\sin(t)\). For \(t = 6\pi\), since \(\sin(6\pi) = 0\), the cosecant function is undefined as division by zero is not allowed.\nThe secant function, \(\sec(t)\), is the reciprocal of the cosine function, so \(\sec(t) = 1/\cos(t)\). For \(t = 6\pi\), since \(\cos(6\pi) = 1\), we have \(\sec(6\pi) = 1/1 = 1\).\nThe cotangent function, \(\cot(t)\), is the reciprocal of the tangent function, so \(\cot(t) = 1/\tan(t)\). For \(t = 6\pi\), since \(\tan(6\pi) = 0\), the cotangent function is undefined as division by zero is not allowed.

Key Concepts

Cosine

The cosine function, often notated as \( \cos(t) \), measures the horizontal coordinate of a point on the unit circle corresponding to the angle \( t \). One of the essential properties of cosine is its periodicity. This means that every \( 2\pi \), the values of cosine repeat.
When analyzing \( \cos(t) \) for values like \( t = 6\pi \), it's important to recognize that \( 6\pi \) is simply 3 full rotations around the circle (because \( 6\pi = 3 \times 2\pi \)).
In trigonometry, any angle that is an integer multiple of \( 2\pi \) will have a cosine value of 1.

• This is because the point comes back to its starting position on the unit circle, which is (1,0).
• Therefore, for the angle \( t = 6\pi \), \( \cos(6\pi) = 1 \).

Sine

The sine function, denoted as \( \sin(t) \), relates to the vertical component of the unit circle corresponding to the angle \( t \). Like cosine, sine is periodic with a period of \( 2\pi \), repeating its pattern over every complete rotation around the unit circle.
To determine \( \sin(t) \) at specific angles, observe the simple oscillation pattern over a full cycle:

• Starts at 0, peaks at 1, returns to 0, dips to -1, and finally returns to 0 as it completes \( 2\pi \).

At \( t = 6\pi \), which again is three complete rotations, the sine value returns to the starting point.
This means:

• \( \sin(6\pi) = 0 \).

Tangent

The tangent function, represented by \( \tan(t) \), can be thought of as the ratio of sine to cosine.It is an important function in trigonometry, as it describes the gradient or slope of the line at any given point on the circle.
To find the value of tangent at \( t = 6\pi \), we apply the definition:\( \tan(t) = \frac{\sin(t)}{\cos(t)} \).
Knowing that \( \sin(6\pi) = 0 \) and \( \cos(6\pi) = 1 \), the calculation becomes:

• \( \tan(6\pi) = \frac{0}{1} = 0 \).

The tangent function is also periodic, but with a different period of \( \pi \), meaning its values repeat every half-circle rotation.