Problem 55 - 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 \(s\). $$5.5$$

Step-by-Step Solution

Answer

\( \cos(5.5) \approx 0.708669 \) and \( \sin(5.5) \approx -0.705541 \); fourth quadrant.

Step 1: Determine the Angle's Radian Measure

The problem specifies the angle in radians as \( s = 5.5 \). Since all calculations will be based on this measure, there's no modification needed at this step.

Step 2: Calculate Cosine and Sine

Use a calculator in radian mode to find the values of \( \cos(5.5) \) and \( \sin(5.5) \). When calculated, \( \cos(5.5) \approx 0.708669 \) and \( \sin(5.5) \approx -0.705541 \).

Step 3: Determine the Quadrant

Since the cosine is positive and the sine is negative, the point \((\cos(5.5), \sin(5.5))\) lies in the fourth quadrant of the unit circle.

Step 4: Calculate Tangent and Cotangent

Use the basic trigonometric identities \( \tan(s) = \frac{\sin(s)}{\cos(s)} \) and \( \cot(s) = \frac{\cos(s)}{\sin(s)} \). For \( s = 5.5 \), calculate \( \tan(5.5) \approx \frac{-0.705541}{0.708669} \approx -0.995586 \) and \( \cot(5.5) \approx \frac{0.708669}{-0.705541} \approx -1.004427 \).

Step 5: Calculate Secant and Cosecant

Use the identities \( \sec(s) = \frac{1}{\cos(s)} \) and \( \csc(s) = \frac{1}{\sin(s)} \). For \( s = 5.5 \), find \( \sec(5.5) \approx \frac{1}{0.708669} \approx 1.410279 \) and \( \csc(5.5) \approx \frac{1}{-0.705541} \approx -1.416952 \).

Key Concepts

Unit Circle

The unit circle is an essential concept in trigonometry. It's a circle with a radius of 1 centered at the origin of a coordinate system. This simple geometric shape allows us to define the trigonometric functions sine, cosine, and tangent using the coordinates of a point on the circle.

- **Quadrants on the Unit Circle**
The circle is divided into four quadrants, each relating to specific ranges of angle measures.

• Quadrant I: both cosine and sine values are positive.
• Quadrant II: cosine is negative, sine is positive.
• Quadrant III: both values are negative.
• Quadrant IV: cosine is positive, sine is negative.

Understanding these quadrants helps to predict and check the signs of trigonometric function values. By knowing where an angle lies on the unit circle, students can reason out the general behavior of trigonometric functions. For instance, knowing that an angle of 5.5 radians is in the fourth quadrant immediately suggests that cosine would be positive while sine would be negative.

Radian Measure

Radian measure is a way of expressing angles, which is critical in trigonometry for its simplicity and natural relation to the unit circle. A radian is defined as the angle created at the center of a circle by an arc whose length is equal to the circle's radius.

- **Understanding Radians**
Using radians is beneficial because it simplifies many formulas in mathematics. Instead of using degrees, which compartmentalize a circle into 360 parts, radians divide a circle into approximately 6.283 parts, which is approximately equal to 2π.

• Half a circle is measured at π radians.
• A full circle is 2π radians.
• One quarter of a circle is π/2 radians.

Working with radians is more intuitive when dealing with calculus and advanced mathematics. When calculating the trigonometric functions of 5.5 radians, you stay consistent with this natural measure, effortlessly finding out that this value fits into the fourth quadrant of the unit circle.

Trigonometric Identities

Trigonometric identities are formulas that provide crucial relationships between the different trigonometric functions. These identities are useful for simplifying expressions and solving trigonometric equations.

- **Key Identities**
Among the most used identities are:

• **Pythagorean Identity**: \( \sin^2(s) + \cos^2(s) = 1 \)
• **Reciprocal Identities**:
  • \( \csc(s) = \frac{1}{\sin(s)} \)
  • \( \sec(s) = \frac{1}{\cos(s)} \)
  • \( \cot(s) = \frac{1}{\tan(s)} \)
• **Quotient Identities**:
  • \( \tan(s) = \frac{\sin(s)}{\cos(s)} \)
  • \( \cot(s) = \frac{\cos(s)}{\sin(s)} \)

These identities become handy in many trigonometric problems, including calculating tangent, cotangent, secant, and cosecant. In the exercise, you can use these identities to find the function values for an angle of 5.5 radians, ensuring each function's properties align with the unit circle's nature and the angle's quadrant.