Problem 64 - Precalculus : Building Concepts and Connections

Question

Fill in the given table with the missing information. Approximate all nonexact anstoers to four decimal places. $$\begin{array}{|c|c|c|c|c|} \hline \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline II & & \ & -1 \\ \hline \end{array}$$

Step-by-Step Solution

Answer

The updated table is: \[ \begin{array}{|c|c|c|c|c|} \hline \text { Quadrant } & \sin t & \cos t & \tan t \\ \hline II & 0.7071 & -0.7071 & -1 \\ \hline \end{array} \]

Step 1: Identify the angle in the second quadrant with \(\tan t = -1\)

From the table, \(\tan t = -1\) is provided. The angles in the second quadrant with \(\tan t = -1\) are those of the form \(\pi - \frac{\pi}{4} = \frac{3\pi}{4}\) and other multiples of \(\pi\). Here, the angle whose tan is -1 in quadrant II is \(\frac{3\pi}{4}\). Therefore, \(t = \frac{3\pi}{4}\).

Step 2: Calculate the sine and cosine

Now that the angle is known, calculate the sine and cosine. In quadrant II, We know that \(\sin t > 0\) and \(\cos t < 0\), this is a property of the trigonometric functions in the plane. The \(\sin t\) and \(\cos t\) for \(t = \frac{3\pi}{4}\) corresponds to \(\frac{\sqrt{2}}{2}\) and \(-\frac{\sqrt{2}}{2}\) respectively.

Step 3: Round to four decimal places

Finally, to approximate the answers to four decimal places, \(\sin t\) becomes \(0.7071\) and \(\cos t\) becomes \(-0.7071\) .

Key Concepts

Quadrants

Understanding the concept of quadrants is essential in trigonometry. The coordinate plane is divided into four sections, known as quadrants, which are numbered counterclockwise starting from the top right. Each quadrant has unique characteristics regarding the signs of trigonometric functions:

• Quadrant I: Both sine and cosine values are positive.
• Quadrant II: Sine is positive, but cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Cosine is positive, but sine is negative.

Identifying these quadrants helps in understanding how the values of trigonometric functions change with angles. For example, in this exercise, the angle with a tangent of -1 falls in Quadrant II where the angle is \(\frac{3\pi}{4}\). This understanding of quadrants allows us to interpret and solve trigonometric problems effectively.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions used to describe the relationships of angles in a right triangle. These functions also extend to the unit circle, which is a powerful tool for understanding trigonometry in all quadrants.
- **Sine (\(\sin t\))** is defined as the y-coordinate of the point on the unit circle, hence it provides the vertical component of the angle.- **Cosine (\(\cos t\))** is the x-coordinate, representing the horizontal component.In the second quadrant, the sine value remains positive while the cosine value becomes negative. For the angle \(t = \frac{3\pi}{4}\), both sine and cosine can be determined by symmetry from known angles in the first quadrant. Therefore, \(\sin t = \frac{\sqrt{2}}{2}\) and \(\cos t = -\frac{\sqrt{2}}{2}\), which approximate to 0.7071 and -0.7071 respectively when rounded to four decimal places.

Tangent

The tangent function is a bit different from sine and cosine as it represents the ratio of the two. It is defined as:\[ \tan t = \frac{\sin t}{\cos t} \]This means that tangent gives us the slope of the line formed by the angle.In Quadrant II, the tangent is negative, which aligns with our exercise, where \(\tan t = -1\). This tells us that the angle is equidistant from the x and y axes, specifically at \(\frac{3\pi}{4}\).Tangent is particularly useful because it helps indicate the steepness of the angle and changes its sign in different quadrants:

• In Quadrant I, tangent is positive.
• In Quadrant II, tangent becomes negative.
• In Quadrant III, tangent is positive again.
• In Quadrant IV, it turns negative.

Knowing these properties helps in predicting the behavior of trigonometric values as angles move between different quadrants.