Problem 25 - Precalculus: A Concise Course

Question

Find the values of the six trigonometric functions of \(\boldsymbol{\theta}\) with the given constraint. $$ \sin \theta=\frac{3}{5} \quad \theta \text { lies in Quadrant II. } $$

Step-by-Step Solution

Answer

The six trigonometric functions for the angle \(\theta\), where \(\sin(\theta) = \frac{3}{5}\) and \(\theta\) lies in Quadrant II, are: \(\sin(\theta) = \frac{3}{5}\), \(\cos(\theta) = -\frac{4}{5}\), \(\tan(\theta) = -\frac{3}{4}\), \(\csc(\theta) = \frac{5}{3}\), \(\sec(\theta) = -\frac{5}{4}\), and \(\cot(\theta) = -\frac{4}{3}\).

Step 1: Find cosine using Pythagorean identity

We can find the cosine of angle \(\theta\) using the Pythagorean identity \(\sin^2(\theta) + \cos^2(\theta) = 1\). Because sine of \(\theta\) is given as \(\frac{3}{5}\), we can substitute that in to the identity to solve for cosine: \(1 - \sin^2(\theta) = \cos^2(\theta)\)Substituting \(\sin(\theta) = \frac{3}{5}\) into the equation, we get:\(\cos^2(\theta) = 1 - (\frac{3}{5})^2 = 1 - \frac{9}{25} = \frac{16}{25}\),Following the principles of the trigonometric functions, since \(\theta\) is in Quadrant II, cosine is negative, hence \(\cos(\theta) = -\sqrt{\frac{16}{25}} = -\frac{4}{5}\).

Step 2: Find tangent using quotient identity

Tangent is the ratio of sine to cosine. Use the quotient identity \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). Substituting the values of sine and cosine of angle \(\theta\) into the formula, we get:\(\tan(\theta) = -\frac{3}{4}\).

Step 3: Find cosecant, secant, and cotangent

Cosecant is the reciprocal of sine, secant is the reciprocal of cosine, and cotangent is the reciprocal of tangent. Therefore:\(\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{3}\),\(\sec(\theta) = \frac{1}{\cos(\theta)} = -\frac{5}{4}\),\(\cot(\theta) = \frac{1}{\tan(\theta)} = -\frac{4}{3}\).

Key Concepts

Quadrant II Trigonometry

In trigonometry, the coordinate plane is divided into four sections, known as quadrants. The angles in Quadrant II range from 90 to 180 degrees, or \frac{\textstyle \text{π}}{\textstyle 2}$ to π in radians. One of the key things to remember about Quadrant II is that the sine values (y-coordinates) are positive, while cosine values (x-coordinates) are negative. This is due to the fact that in this quadrant, the terminal side of the angle lies in the space where the x values are negative and the y values are positive.

When solving for trigonometric functions in Quadrant II, it's crucial to take the sign of the functions into account. For example, with \(sin \theta = \frac{3}{5}\) in Quadrant II, we find cosine to be negative, which influences the signs of related functions such as tangent and secant. Understanding these quadrant-specific characteristics is essential for accurately determining the values of all trigonometric functions.

Pythagorean Identity

Understanding the Pythagorean Identity

One of the fundamental ideas in trigonometry is the Pythagorean identity, which states that for any angle θ, \(sin^2(\theta) + cos^2(\theta) = 1\). This identity is derived from the Pythagorean theorem related to the sides of a right triangle. In the context of a unit circle, where the radius is 1, the identity depicts the relationship between the projections on the x and y axes, which are the cosine and sine values, respectively.

Applying the Pythagorean Identity

To find the cosine of an angle when the sine is known (like in our exercise with \(sin \theta = \frac{3}{5}\)), you subtract the square of the sine from 1 and then take the square root to find the cosine value. Remember to consider the sign based on the quadrant. In Quadrant II, cosine must be negative, leading to a cosine value of \( -\frac{4}{5}\) for our given angle. This process demonstrates the utility of the Pythagorean identity in finding one trigonometric function from another.

Quotient Identity

Exploring the Quotient Identity

Trigonometry involves several identities, with the quotient identity being particularly useful. This identity expresses the tangent function as the ratio of sine to cosine: \(tan(\theta) = \frac{sin(\theta)}{cos(\theta)}\). The quotient identity underlines the interconnected nature of trigonometric functions.

Using the Quotient Identity in Practice

In our exercise, we find that \(tan \theta = \frac{sin \theta}{cos \theta}\) leads to \(tan \theta = \frac{3/5}{-4/5}\), simplifying to \(tan \theta = -\frac{3}{4}\). It’s essential to remember to carry the sign through the calculation, as it indicates the function's direction in the specific quadrant. In this case, as theta is in Quadrant II, where both sine is positive and cosine is negative, the tangent value will be negative. Clearly understanding and applying the quotient identity can help solve various trigonometric problems.