Problem 57 - Contemporary Precalculus

Question

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\cos t=0 \quad \text { and } \quad \sin t=1$$

Step-by-Step Solution

Answer

Answer: In this case, the values of the six trigonometric functions are as follows: \(\sin t = 1\\ \cos t = 0\\ \tan t = \text{undefined}\\ \csc t = 1\\ \sec t = \text{undefined}\\ \cot t = 0\)

Step 1: Recap basic trigonometric relationships

Using the given conditions \(\cos t = 0\) and \(\sin t = 1\), we can find the remaining trigonometric functions by using their basic definitions and relationships: $$\tan t = \frac{\sin t}{\cos t}$$ $$\csc t = \frac{1}{\sin t}$$ $$\sec t = \frac{1}{\cos t}$$ $$\cot t = \frac{\cos t}{\sin t}$$

Step 2: Calculate the tangent function

We can find the tangent function by substituting the given values for cosine and sine: $$\tan t = \frac{\sin t}{\cos t} = \frac{1}{0}$$ Since we cannot divide by zero, we note that the tangent function is undefined in this case.

Step 3: Calculate the cosecant function

We can find the cosecant function using its definition: $$\csc t = \frac{1}{\sin t} = \frac{1}{1} = 1$$

Step 4: Calculate the secant function

Similarly, we can find the secant function: $$\sec t = \frac{1}{\cos t} = \frac{1}{0}$$ Since we cannot divide by zero, we note that the secant function is undefined in this case.

Step 5: Calculate the cotangent function

Finally, we can find the cotangent function: $$\cot t = \frac{\cos t}{\sin t} = \frac{0}{1} = 0$$

Step 6: Summarize the results

In conclusion, the values of the six trigonometric functions for the given conditions are: \(\sin t = 1\\ \cos t = 0\\ \tan t = \text{undefined}\\ \csc t = 1\\ \sec t = \text{undefined}\\ \cot t = 0\)

Key Concepts

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to each other. They allow us to express one trigonometric function in terms of another, making it easier to solve equations or compute values when given a specific condition. In the given exercise, knowing these identities helps in finding other trigonometric functions when \( \cos t = 0 \) and \( \sin t = 1 \).

Some essential trigonometric identities include:

• \( \tan t = \frac{\sin t}{\cos t} \)
• \( \csc t = \frac{1}{\sin t} \)
• \( \sec t = \frac{1}{\cos t} \)
• \( \cot t = \frac{\cos t}{\sin t} \)

By applying these identities, you can determine all trigonometric functions based on the given sine and cosine values.

Sine Function

The sine function, abbreviated as \( \sin \), is one of the most important trigonometric functions. It is defined as the ratio of the length of the side opposite to the angle in a right triangle to the hypotenuse. In the unit circle, the sine of an angle is the y-coordinate of the corresponding point.

For our exercise condition \( \sin t = 1 \), this indicates two possible scenarios:

• On the unit circle, the angle \( t \) is \( \frac{\pi}{2} \) radians or \( 90^{\circ} \).
• This also means that at this angle, the corresponding point is directly at the top of the unit circle.

This position results in several effects on other trigonometric functions, such as making the cosine zero.

Cosine Function

The cosine function, written as \( \cos \), is another vital trigonometric function, defined as the ratio of the adjacent side to the hypotenuse of a right triangle. In the unit circle framework, \( \cos \) represents the x-coordinate of a point on the circle.

Given \( \cos t = 0 \) in the exercise, the angle \( t \) positions the point right at the top or bottom of the unit circle. This gives two possible angles:

• \( \frac{\pi}{2} \) radians (90 degrees), where \( \sin t = 1 \)
• \( \frac{3\pi}{2} \) radians (270 degrees), where \( \sin t = -1 \)

In our specific scenario, \( \sin t = 1 \), confirming our angle as \( \frac{\pi}{2} \) radians.

Tangent Function

The tangent function, expressed as \( \tan \), is defined as the ratio of the sine of an angle to its cosine. Mathematically, this is written as \( \tan t = \frac{\sin t}{\cos t} \).

In exercises like this, checking the tangent is important because:

• When \( \cos t = 0 \), \( \tan t \) becomes undefined as division by zero is not allowed.
• Thus, the tangent function tells you information about critical points where an angle may be vertical (like straight up or down on the circle)

In our case, with \( \cos t = 0 \) and \( \sin t = 1 \), the tangent function is undefined.

Cosecant Function

The cosecant function, denoted as \( \csc \), is the reciprocal of the sine function. This means \( \csc t = \frac{1}{\sin t} \). It becomes very straightforward to evaluate once the sine of an angle is known.

In the given exercise, since \( \sin t = 1 \), calculating the cosecant is easy:

• \( \csc t = \frac{1}{1} = 1 \)
• This reciprocal relationship underscores the connection between the sine and cosecant functions, indicating that wherever the sine is maximum, so is the cosecant.

This makes the cosecant simple to determine without confusion or complications.

Secant Function

The secant function, written as \( \sec \), is the reciprocal of the cosine function and defined as \( \sec t = \frac{1}{\cos t} \). Evaluating the secant gives insight into angles where the cosine may be zero.

In this exercise, since \( \cos t = 0 \), the secant function becomes undefined:

• This occurs because division by zero is not possible.
• It reflects the angle where the cosine rotates the point to the top of the unit circle.

The secant indicates crucial vertical asymptote points on trigonometric graphs due to this behavior.

Cotangent Function

The cotangent function, abbreviated as \( \cot \), is the reciprocal of the tangent function, meaning \( \cot t = \frac{\cos t}{\sin t} \). Understanding cotangent aids in analyzing certain symmetries in angles.

In our scenario, because \( \sin t = 1 \) and \( \cos t = 0 \), the calculation simplifies:

• \( \cot t = \frac{0}{1} = 0 \)
• This highlights that when the sine value dominates (at maxima), the cotangent approaches zero.

These observations can strategically simplify your trigonometric problem-solving by knowing when the cotangent reaches zero.