Problem 50 - 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 \(s\). $$0.95$$

Step-by-Step Solution

Answer

Cosine and sine are approximately 0.5820 and 0.8134; the angle is in the first quadrant.

Step 1: Calculate cos(0.95)

Set your calculator to radian mode, and calculate the cosine of 0.95. Enter the value 0.95 into your calculator as the angle and use the cosine function. You should get approximately \( ext{cos}(0.95) \approx 0.5820\).

Step 2: Calculate sin(0.95)

Use the radian mode on your calculator to compute the sine of 0.95. Enter 0.95 as the angle and use the sine function. This will give you approximately \( ext{sin}(0.95) \approx 0.8134\).

Step 3: Determine the quadrant

Since both \(\text{cos}(0.95)\) and \(\text{sin}(0.95)\) are positive, the angle 0.95 is in the first quadrant because both cosine and sine are positive in the first quadrant.

Step 4: Calculate tan(0.95)

The tangent of an angle is given by \(\text{tan}(s) = \frac{\text{sin}(s)}{\text{cos}(s)}\). Thus, \(\text{tan}(0.95) = \frac{0.8134}{0.5820} \approx 1.3971\).

Step 5: Calculate cot(0.95)

The cotangent is the reciprocal of the tangent: \(\text{cot}(s) = \frac{1}{\text{tan}(s)}\). Calculate \(\text{cot}(0.95) \approx \frac{1}{1.3971} \approx 0.7159\).

Step 6: Calculate sec(0.95)

The secant is the reciprocal of the cosine: \(\text{sec}(s) = \frac{1}{\text{cos}(s)}\). Therefore, \(\text{sec}(0.95) \approx \frac{1}{0.5820} \approx 1.7187\).

Step 7: Calculate csc(0.95)

The cosecant is the reciprocal of the sine: \(\text{csc}(s) = \frac{1}{\text{sin}(s)}\). So, \(\text{csc}(0.95) \approx \frac{1}{0.8134} \approx 1.2292\).

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry, providing insight into the relationships between angles and the lengths of chords. Imagine a circle centered at the origin of a Cartesian plane with a radius of one unit. Every point on this circle has coordinates reflecting trigonometric values.

• The x-coordinate of a point on the unit circle represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

Understanding how angles and coordinates interact on the unit circle helps manage calculations involving trigonometric functions.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent play a vital role in connecting angles to side ratios in right triangles. These functions are also used to describe circular motion and periodic phenomena.

• Sine \(\sin(\theta)\): This function returns the y-coordinate of the point on the unit circle.
• Cosine \(\cos(\theta)\): This gives the x-coordinate of the relevant point on the unit circle.
• Tangent \(\tan(\theta)\): Defined as the ratio of sine to cosine, it represents the slope of the line that intersects the terminal side of the angle on the unit circle.

These fundamental trigonometric functions form the basis for more complex calculations such as cotangent, secant, and cosecant which are simply their respective reciprocals.

Radian Mode

Radian mode is critical for interpreting angles within the unit circle framework. Unlike degrees, radians provide a natural measure of angles, using the arc length of a circle divided by its radius.

• One full circle is equivalent to \(2\pi\) radians.
• Half a circle is \(\pi\) radians.
• An angle of \((\pi/2)\) radians describes a quarter of the unit circle.

Setting your calculator to radian mode ensures that calculations of trigonometric functions reflect these natural circle partitions, which is essential in many scientific and engineering fields.

Quadrants

The unit circle is divided into four quadrants, each representing a unique combination of positive or negative sine and cosine values.

• First Quadrant (0 to \(\pi/2\)): Both sine and cosine are positive.
• Second Quadrant (\((\pi/2)\) to \(\pi\)): Sine is positive, cosine is negative.
• Third Quadrant (\(\pi\) to \(3\pi/2\)): Both sine and cosine are negative.
• Fourth Quadrant (\((3\pi/2)\) to \(2\pi\)): Sine is negative, cosine is positive.

When interpreting angles like 0.95 radians, identifying the quadrant helps determine the signs of their trigonometric values and thus interpreting the overall geometry of the problem.