Problem 56 - 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 \(s\). $$5.75$$

Step-by-Step Solution

Answer

\(\cos(5.75) \approx 0.8522\), \(\sin(5.75) \approx -0.5234\), Quadrant IV; \(\tan(5.75) \approx -0.6142\), \(\cot(5.75) \approx -1.6283\), \(\sec(5.75) \approx 1.1732\), \(\csc(5.75) \approx -1.9104\).

Step 1: Calculate Cosine and Sine

In radian mode, input the value 5.75 into your calculator to find \(\cos(5.75)\) and \(\sin(5.75)\). The calculator gives the values:\[\cos(5.75) \approx 0.8522\] \[\sin(5.75) \approx -0.5234\]

Step 2: Determine the Quadrant

To determine where \(s = 5.75\) radians lies on the unit circle, we consider the positive sine and cosine values. When \(\cos(s) > 0\) and \(\sin(s) < 0\), point \(s\) is in the fourth quadrant.

Step 3: Calculate Tangent

Using the trigonometric identity \(\tan(s) = \frac{\sin(s)}{\cos(s)}\), find the tangent value:\[\tan(5.75) \approx \frac{-0.5234}{0.8522} \approx -0.6142\]

Step 4: Calculate Cotangent

Using the reciprocal identity \(\cot(s) = \frac{1}{\tan(s)}\), find the cotangent value:\[\cot(5.75) \approx \frac{1}{-0.6142} \approx -1.6283\]

Step 5: Calculate Secant

Using the reciprocal identity \(\sec(s) = \frac{1}{\cos(s)}\), find the secant value:\[\sec(5.75) \approx \frac{1}{0.8522} \approx 1.1732\]

Step 6: Calculate Cosecant

Using the reciprocal identity \(\csc(s) = \frac{1}{\sin(s)}\), find the cosecant value:\[\csc(5.75) \approx \frac{1}{-0.5234} \approx -1.9104\]

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. Picture a circle centered at the origin of a coordinate plane with a radius of 1. This simple geometric shape plays a crucial role in defining trigonometric functions for angles. Each point on the unit circle corresponds to the cosine and sine of an angle.When you plot an angle, typically measured in radians, from the positive x-axis around the circle, the x-coordinate of the point where the angle's terminal side intersects the circle is the cosine of the angle. Similarly, the y-coordinate is the sine of the angle. These values are incredibly useful in calculating other trigonometric functions: tangent, cotangent, secant, and cosecant.The unit circle is divided into four quadrants:

• First Quadrant: \(0 \leq \theta \leq \frac{\pi}{2}\), both sine and cosine are positive.
• Second Quadrant: \(\frac{\pi}{2} \leq \theta \leq \pi\), sine is positive, cosine is negative.
• Third Quadrant: \(\pi \leq \theta \leq \frac{3\pi}{2}\), both sine and cosine are negative.
• Fourth Quadrant: \(\frac{3\pi}{2} \leq \theta \leq 2\pi\), cosine is positive, sine is negative.

Understanding which quadrant an angle lies in can help predict the signs of the trigonometric functions, especially when using a calculator for approximations.

Radian Mode

Radians are another way to measure angles, aside from degrees. One full revolution around a circle, which is 360 degrees, is equivalent to \(2\pi\) radians. Hence, \pi \, radians is 180 degrees, etc. Radians offer a more natural and mathematical approach to angle measurement because they are based on the radius of the circle.When using a calculator to solve trigonometric problems that involve radians, it is important to set the calculator to "radian mode." This ensures calculations are made under the assumption that inputs are in radians rather than degrees, which would lead to incorrect trigonometric values otherwise.For example:

• To find \(\cos(5.75)\), ensure your calculator is in radian mode, or else it might calculate the cosine of 5.75 degrees instead.
• When calculating tangent as \(\tan(s) = \frac{\sin(s)}{\cos(s)}\), remember that both sine and cosine should be derived with angles in radian measure.

By adopting radian measure for calculations, mathematicians and scientists often find their work integrating more seamlessly with calculus and other advanced mathematical studies.

Trigonometric Identities

Trigonometric identities are equations that establish relationships between the trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. These identities are essential tools for solving complex trigonometric equations and proving mathematical results.Some common identities include:

• Pythagorean Identity: \sin^2(\theta) + \cos^2(\theta) = 1\
• Tangent and Cotangent: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\), \(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)
• Reciprocal Identities: \(\csc(\theta) = \frac{1}{\sin(\theta)}\), \(\sec(\theta) = \frac{1}{\cos(\theta)}\)

When calculating the tangent, cotangent, secant, or cosecant for a given angle \(s\) like 5.75, these identities become instrumental. For instance, if you have the sine and cosine values, you can determine tangent as \(\tan(s) = \frac{\sin(s)}{\cos(s)}\), which was approximately \(-0.6142\) for 5.75.These relationships not only simplify complex calculations, but also help in understanding how different trigonometric values change as angles vary. Knowing them thoroughly allows for deeper insight into the workings of trigonometry and calculus.