Problem 25 - Algebra and Trigonometry

Question

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. \(t=0\)

Step-by-Step Solution

Answer

\( \sin(0) = 0, \cos(0) = 1, \tan(0) = 0, \csc(0) = \text{undefined}, \sec(0) = 1, \cot(0) = \text{undefined} \).

Step 1: Sine Function

Start by finding \( \sin(t) \) where \( t = 0 \). The sine of 0 radians is 0. Therefore, \( \sin(0) = 0 \).

Step 2: Cosine Function

Next, determine \( \cos(t) \) where \( t = 0 \). The cosine of 0 radians is 1. Hence, \( \cos(0) = 1 \).

Step 3: Tangent Function

Now, let's find \( \tan(t) \), which is the ratio of sine to cosine. So, \( \tan(t) = \frac{\sin(t)}{\cos(t)} = \frac{0}{1} = 0 \). Therefore, \( \tan(0) = 0 \).

Step 4: Cosecant Function

For \( \csc(t) \), find the reciprocal of sine. Since \( \sin(0) = 0 \), \( \csc(0) = \frac{1}{0} \). This is undefined because division by zero is not possible.

Step 5: Secant Function

Next, determine \( \sec(t) \), which is the reciprocal of cosine. Since \( \cos(0) = 1 \), \( \sec(0) = \frac{1}{1} = 1 \).

Step 6: Cotangent Function

Finally, find \( \cot(t) \), the reciprocal of tangent. Since \( \tan(0) = 0 \), \( \cot(0) = \frac{1}{0} \). This is also undefined due to division by zero.

Key Concepts

Sine Function

The sine function, often represented by \( \sin(t) \), measures how high up or down a point is on the unit circle at an angle \( t \). Think of the unit circle as a hula hoop lying on the ground, with you standing in the middle looking up. The angle \( t \) measures how far you've turned from the starting point, right along the hoop's edge.

• When \( t = 0 \), you're right at the starting point, where the unit circle in mathematics has all points at height 0.
• Thus, \( \sin(0) = 0 \).

In any angle, the sine value will tell you how far up or down the point is from the horizontal line through the center. The range of the sine function lies between -1 and 1, making it easy to remember its highest and lowest values achieved at specific angles like 90° or 270° (or equivalently \( \pi/2 \) or \( 3\pi/2 \) radians).

Cosine Function

The cosine function, denoted by \( \cos(t) \), measures the horizontal distance from the center of the unit circle to the point along the edge of the circle, when you turn the full angle \( t \).

• At \( t = 0 \), the cosine value gives the rightmost point on the circle, which is precisely one unit away from the center.
• Hence, \( \cos(0) = 1 \).

The cosine function helps indicate how far left or right your position is compared to the vertical axis. Its value ranges from -1 to 1, similar to sine, but with the peaks occurring at different angles like 0° or 360° (or \( 0 \) and \( 2\pi \) radians). The neat symmetry of these functions means when sine is zero at a specific angle, cosine is often at a value of 1 or -1, and vice versa.

Tangent Function

The tangent function, shown as \( \tan(t) \), gives you the slope of the line that touches just one spot on the unit circle—the tangent line. This slope is the ratio of the sine value (vertical component) to the cosine value (horizontal component) at a given angle.

• At \( t = 0 \), this ratio becomes \( \frac{\sin(0)}{\cos(0)} = \frac{0}{1} = 0 \).
• Therefore, \( \tan(0) = 0 \).

Tangent values can be anything from negative to positive infinity, because while the sine and cosine range between -1 and 1, their division can shoot the value of tangent way beyond those limits. Whenever cosine becomes zero, tangent scores big, being undefined since we're unable to divide by zero. This characteristic gives the tangent function its distinct pattern of repeating undefined breaks at every \( \pi/2 \) or \( 3\pi/2 \) radians, creating vertical asymptotes along its graph.