Problem 52 - A Graphical Approach to Precalculus with Limits

Question

Use a calculator in radian mode to find approximations for cos \(s\) and sin s for each number s. Give as many decimal places as your calculator displays. (These are NOT exact values-they are only approximations.) Then determine the quadrant in which the point on the unit circle corresponding to s lies. Finally. find approximations for tan \(s\), cot \(s\), sec \(s\), and cse \(s\). $$-3.75$$

Step-by-Step Solution

Answer

Cosine is approximately -0.820559357, sine is approximately 0.571561318, and the angle is in Quadrant II.

Step 1: Set Calculator to Radian Mode

Ensure that your calculator is set to radian mode. This step is crucial because the value given (-3.75) is in radians. Double-check your calculator's settings to avoid any errors in computation.

Step 2: Calculate Cosine of s

Input the value \(-3.75\) into your calculator and use the cosine function to find \(\cos(-3.75)\). Record the value displayed on your calculator. The approximate value is \(\cos(-3.75) \approx -0.820559357\).

Step 3: Calculate Sine of s

Input the same value \(-3.75\) and use the sine function to find \(\sin(-3.75)\). Record the value displayed.The approximate value is \(\sin(-3.75) \approx 0.571561318\).

Step 4: Determine the Quadrant

Since \(-3.75\) is negative, it indicates a clockwise direction starting from the positive x-axis. The cosine is negative, and the sine is positive. This places \(-3.75\) radians in Quadrant II of the unit circle.

Step 5: Calculate Tangent of s

Tangent is calculated using the ratio of sine to cosine, \(\tan s = \frac{\sin s}{\cos s}\).Substitute the values: \(\tan(-3.75) \approx \frac{0.571561318}{-0.820559357} \approx -0.6967147221\).

Step 6: Calculate Cotangent of s

Cotangent is the reciprocal of tangent, \(\cot s = \frac{1}{\tan s}\). Substitute the calculated tangent:\(\cot(-3.75) \approx \frac{1}{-0.6967147221} \approx -1.43592405\).

Step 7: Calculate Secant of s

Secant is the reciprocal of cosine, \(\sec s = \frac{1}{\cos s}\).Substitute the cosine value:\(\sec(-3.75) \approx \frac{1}{-0.820559357} \approx -1.218978667\).

Step 8: Calculate Cosecant of s

Cosecant is the reciprocal of sine, \(\csc s = \frac{1}{\sin s}\).Substitute the sine value:\(\csc(-3.75) \approx \frac{1}{0.571561318} \approx 1.749001365\).

Key Concepts

Unit Circle

The unit circle is a fundamental tool in trigonometry. It is a circle with a radius of one centered at the origin of a coordinate plane. Every angle on the unit circle can be defined in terms of the coordinates of a point on this circle. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle, while the y-coordinate relates to the sine of the angle. This unique property makes the unit circle a powerful tool for understanding trigonometric functions.

• Coordinates \( ( ext{x}, ext{y}) \) of a point on the unit circle represent \( ( ext{cos} \, \theta , ext{sin} \, \theta ) \)
• Unit circle enables easy visualization of angle measurements
• Helps determine the values of sine, cosine and other trigonometric ratios

Radian Mode

Radians are a way to measure angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians use the circle's radius to measure the angle's arc length. One complete revolution around a circle is equivalent to approximately 6.283 radians, or just \(2\pi\), making them a natural fit for calculations involving circular motion.
In trigonometry, we often use calculators to compute functions like sine or cosine. In these calculations, using radian mode ensures the angle input gives correct results. Therefore, it’s crucial to set your calculator to radian mode whenever dealing with angles given in radians in problems related to the unit circle or trigonometry in general.

Trigonometric Ratios

Trigonometric ratios are functions that relate the angles of a right triangle to the ratios of its sides. There are six primary trig functions: sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These functions are essential for understanding various properties of angles and can be computed using a calculator in radian mode.

• The sine and cosine functions derive directly from an angle's coordinates on the unit circle.
• The tangent function is the ratio of sine to cosine: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
• The reciprocal functions are: \(\csc \theta = \frac{1}{\sin \theta}, \sec \theta = \frac{1}{\cos \theta}, \cot \theta = \frac{1}{\tan \theta}\)

Each ratio provides valuable insights, such as understanding the slope of an angle representing its direction and orientation on a plane.

Quadrants

The coordinate plane is divided into four sections known as quadrants. These quadrants are numbered counterclockwise, starting from the positive x-axis. Each quadrant is associated with specific signs for sine and cosine functions, providing insights into the angle's trigonometric characteristics.

• Quadrant I (0 to \(\frac{\pi}{2}\)) - Sine and cosine are both positive
• Quadrant II (\(\frac{\pi}{2} \) to \pi) - Sine is positive, cosine is negative
• Quadrant III (\(\pi \) to \(\frac{3\pi}{2}\)) - Sine and cosine are both negative
• Quadrant IV (\(\frac{3\pi}{2} \) to \(2\pi\)) - Sine is negative, cosine is positive

These quadrants help determine the trigonometric functions' signs, making it easier to understand an angle's position relative to the unit circle. It’s particularly useful for solving problems involving angles not in standard positions, like negative radian values.