Problem 26 - Precalculus Essentials

Question

Sin \(t\) and cos \(t\) are given. Use identities to find tan \(t,\) cse \(t,\) sec \(t,\) and cot \(t .\) Where necessary, rationalize denominators. $$\sin t=\frac{3}{5}, \cos t=\frac{4}{5}$$

Step-by-Step Solution

Answer

The values of the trigonometric functions are tan \(t\) = \(\frac{3}{4}\), csc \(t\) = \(\frac{5}{3}\), sec \(t\) = \(\frac{5}{4}\), and cot \(t\) = \(\frac{4}{3}\)

Step 1: Compute tan \(t\)

Using the identity tan \(t\) = sin \(t\) / cos \(t\), substituting the given values we get: tan \(t\) = \(\frac{3}{5}\) / \(\frac{4}{5}\) = \(\frac{3}{4}\)

Step 2: Compute csc \(t\)

Using the identity csc \(t\) = 1 / sin \(t\), substituting the given values we get: csc \(t\) = 1 / \(\frac{3}{5}\) = \(\frac{5}{3}\)

Step 3: Compute sec \(t\)

Using the identity sec \(t\) = 1 / cos \(t\), substituting the given values we get: sec \(t\) = 1 / \(\frac{4}{5}\) = \(\frac{5}{4}\)

Step 4: Compute cot \(t\)

Using the identity cot \(t\) = 1 / tan \(t\), substituting the tan \(t\) value obtained in step 1 we get: cot \(t\) = 1 / \(\frac{3}{4}\) = \(\frac{4}{3}\)

Key Concepts

Sine and Cosine

In trigonometry, sine and cosine are foundational concepts used to describe the ratios of sides in a right triangle. The sine of an angle in a right triangle is the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Similarly, the cosine is the ratio of the length of the adjacent side to the hypotenuse. Mathematically, they are expressed as follows:

• ext{ ext{Sine} = rac{ ext{Opposite side}}{ ext{Hypotenuse}} ext{Cosine} = rac{ ext{Adjacent side}}{ ext{Hypotenuse}} }

The values of sine and cosine always range from (-1) to 1. This property is pivotal for their function in oscillations, waves, and circles. In the given exercise, we have ( ext{sin } t = rac{3}{5}) and ( ext{cos } t = rac{4}{5}). These ratios confirm the identity ( ext{sin}^{2} t + ext{cos}^{2} t = 1), a fundamental relation affirming the angle's position on the unit circle. This identity is integral in determining other trigonometric functions.

Tangent and Cotangent

Tangent and cotangent are important trigonometric functions related to sine and cosine. The tangent of an angle is defined as the ratio of sine to cosine. Thus, it helps in understanding the slope of angles in a unit circle. This can be calculated as:

• ext{Tangent (tan) = rac{ ext{Sine}}{ ext{Cosine}}}

Given the exercise values, ( ext{tan } t = rac{3}{5} / rac{4}{5} = rac{3}{4}). Tan 't' describes the gradient of angle t when represented in a Cartesian plane.
Likewise, cotangent is the reciprocal of tangent. It is the ratio of cosine to sine, offering another perspective on the angle's slope:

• ext{Cotangent (cot) = rac{ ext{Cosine}}{ ext{Sine}}}

In our exercise, ( ext{cot } t = rac{4}{5} / rac{3}{5} = rac{4}{3}). This shows the inverse relationship between angle and slope in trigonometric functions.

Cosecant and Secant

Cosecant and secant are the reciprocals of sine and cosine, respectively. These functions are crucial for extending our understanding of trigonometry beyond simple right-angle scenarios, often used in calculus and more advanced fields. Cosecant (csc), being the reciprocal of sine, provides insight into how often a particular angle's sine reaches a value over cycles. It is calculated as:

• ext{Cosecant (csc) = rac{1}{ ext{Sine}}}

For this exercise, ( ext{csc } t = rac{1}{rac{3}{5}} = rac{5}{3}).
Similarly, secant is the reciprocal of cosine. It emphasizes the dependence of an angle's cosine over its range depending on the hypotenuse length:

• ext{Secant (sec) = rac{1}{ ext{Cosine}}}

Thus, ( ext{sec } t = rac{1}{rac{4}{5}} = rac{5}{4}) in the exercise. Understanding these relationships aids in calculating distances in circles and mapping complex waves.