Problem 49 - 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 \(s\). $$0.75$$

Step-by-Step Solution

Answer

Cos: 0.7317, Sin: 0.6816, 1st quadrant, Tan: 0.9316, Cot: 1.0732, Sec: 1.3661, Csc: 1.4664.

Step 1: Set the Calculator to Radian Mode

Ensure that your calculator is set to radian mode since the angle given, 0.75, is in radians. Look for a mode button or setting option on your calculator and select radian mode.

Step 2: Calculate Cosine and Sine

Using your calculator, enter 0.75 and find the cosine value. Similarly, calculate the sine value by entering 0.75 and pressing the sine function.

Step 3: Record Approximations

The cosine of 0.75 radians is approximately 0.7316888689 and the sine is approximately 0.6816387602. These values are calculated using your calculator set in radian mode.

Step 4: Determine Quadrant

Since both cosine and sine values are positive, the angle 0.75 radians lies in the first quadrant. In this quadrant, both the x (cosine) and y (sine) values are positive.

Step 5: Calculate Tangent and Cotangent

The tangent of an angle is given by the sine divided by the cosine. Use your calculator to find the tangent by dividing 0.6816387602 by 0.7316888689. Similarly, find the cotangent by dividing the cosine by the sine.

Step 6: Calculate Secant and Cosecant

The secant is the reciprocal of the cosine, and the cosecant is the reciprocal of the sine. Enter the approximate cosine and sine values into your calculator and calculate their reciprocals to find secant and cosecant, respectively.

Step 7: Record All Trigonometric Functions

After calculations, you should get: \(\tan(0.75) \approx 0.9315964599440728\), \(\cot(0.75) \approx 1.0731673205050082\), \(\sec(0.75) \approx 1.3660853490581254\), and \(\csc(0.75) \approx 1.4664404600548968\).

Key Concepts

Radians

Radians are a measure of angles used commonly in trigonometry. Unlike degrees, which divide a circle into 360 parts, radians are based on the radius of a circle. One full circle is equivalent to \(2\pi\) radians, which is approximately 6.28318. This means that \(1\) radian is about \(57.2958^{\circ}\). Using radians can simplify many mathematical calculations, especially those involving periodic functions like sine and cosine. Radians are the standard unit of angular measure in many fields of mathematics and physics.

It's important to understand that radian measures correlate directly with arc length on the unit circle. When working with angles in calculus and advanced trigonometry, radians provide a natural and consistent measure for defining angles and evaluating trigonometric functions. Remember, when using a calculator for trigonometric calculations involving radians, you must adjust your calculator to radian mode to obtain correct results.

Unit Circle

The unit circle is a fundamental concept in trigonometry indispensable for understanding the behavior of trigonometric functions. It is a circle with a radius of one unit, centered at the origin of a coordinate system. The unit circle allows for the definition of sine, cosine, and tangent in terms of coordinates

On the unit circle:

• The x-coordinate corresponds to the cosine of the angle.
• The y-coordinate corresponds to the sine of the angle.

Understanding the unit circle can help avoid reliance on calculators by allowing visualization of sine and cosine values directly from the circle's geometry. For example, when an angle is given in radians, such as \(0.75\), the point corresponding to it on the unit circle can be used to easily determine the sine and cosine values, as well as other trigonometric functions like tangent or secant.

This framework helps us determine not only the trigonometric values but also the quadrant in which the angle lies, based on the positive or negative values of sine and cosine.

Calculator Use

Using a calculator correctly is vital when dealing with trigonometric functions, especially in terms of setting the correct unit for angles—degrees or radians. Since the exercise involves the angle given in radians, making sure your calculator is in radian mode is crucial to ensuring that you get accurate results. Most calculators have a mode setting, usually accessible through a mode button, where you can toggle between radian and degree measurement.

When calculating trigonometric functions such as sine, cosine, or tangent:

• Enter the angle value, here it would be \(0.75\).
• Select the function you want to calculate (e.g., sin, cos, tan).

After calculating the main functions, you can deduce the other functions like cotangent, secant, and cosecant as well, by using reciprocal relations:

• Cotangent is the reciprocal of tangent, secant is the reciprocal of cosine, and cosecant is the reciprocal of sine.

By understanding how to correctly set and use your calculator, you can obtain precise approximations for all trigonometric functions.

Quadrants

The coordinate system is divided into four sections, known as quadrants, which are essential for determining the sign and sometimes the approximate values of trigonometric functions. Starting from the positive x-axis and moving counterclockwise, the quadrants are numbered as follows:

• First Quadrant: Both sine and cosine are positive.
• Second Quadrant: Sine is positive, cosine is negative.
• Third Quadrant: Both sine and cosine are negative.
• Fourth Quadrant: Sine is negative, cosine is positive.

When determining the quadrant for any given angle in radians, you can understand which quadrant the angle falls into by analyzing the signs of the sine and cosine values. For example, for an angle of \(0.75\) radians, where both sine and cosine values are positive, this indicates that the angle lies in the first quadrant. Recognizing the quadrant can help when predicting the behavior of the angle and further related calculations.