Problem 7 - Thomas Calculus

Question

In Exercises \(7-12,\) one of \(\sin x, \cos x,\) and \(\tan x\) is given. Find the other two if \(x\) lies in the specified interval. $$ \sin x=\frac{3}{5}, \quad x \in\left[\frac{\pi}{2}, \pi\right] $$

Step-by-Step Solution

Answer

\( \cos x = -\frac{4}{5}\), \( \tan x = -\frac{3}{4}\).

Step 1: Identify the Quadrant

Given that \( x \) lies in the interval \( \left[\frac{\pi}{2}, \pi\right] \), this means \( x \) is in the second quadrant where sine is positive and cosine and tangent are negative.

Step 2: Determine the Pythagorean Identity

Recall the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). With \( \sin x = \frac{3}{5} \), we can substitute this into the identity to find \( \cos x \).

Step 3: Solve for \( \cos x \)

Substitute \( \sin x = \frac{3}{5} \) into the identity:\[ \left(\frac{3}{5}\right)^2 + \cos^2 x = 1 \]\[ \frac{9}{25} + \cos^2 x = 1 \]\[ \cos^2 x = 1 - \frac{9}{25} = \frac{16}{25} \]Since \( \cos x \) is negative in the second quadrant, \( \cos x = -\frac{4}{5} \).

Step 4: Compute \( \tan x \)

The tangent function is defined as the ratio of sine to cosine: \( \tan x = \frac{\sin x}{\cos x} \). Substitute the values:\[ \tan x = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4} \]

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of sine and cosine functions. This identity is given by the equation:\[\sin^2 x + \cos^2 x = 1\] This is similar to the famous Pythagorean theorem in geometry, and it helps us to find unknown trigonometric values when limited information is available. For instance, if you know \(\sin x\), you can find \(\cos x\) using this identity and vice versa. In the example, we know that \(\sin x = \frac{3}{5}\). Plugging this into the identity, we computed \(\cos^2 x\):\[\left(\frac{3}{5}\right)^2 + \cos^2 x = 1\]Thus, \(\cos^2 x\) can be calculated, and solving,\[\cos^2 x = \frac{16}{25}\]This shows the power of the Pythagorean Identity in solving trigonometric problems, especially when the angle \( x \) is not directly known.

Second Quadrant

When dealing with trigonometric functions and their signs, recognizing the quadrant where an angle lies is crucial. The problem specifies that angle \( x \) is in the Second Quadrant, which is the interval \(\left[\frac{\pi}{2}, \pi\right]\), or between 90° and 180°. In this quadrant, the sine function is positive because we are still above the x-axis compared to the first quadrant while the cosine and tangent are negative because we are now on the left side of the y-axis. This affects the sign of trigonometric ratios:

• \(\sin x\) is positive.
• \(\cos x\) is negative.
• \(\tan x\), being the ratio of \(\sin x\) to \(\cos x\), is negative.

Knowing which functions are positive or negative in each quadrant allows us to correctly determine the sign of any computed trigonometric values, such as when we found \(\cos x = -\frac{4}{5}\) and \(\tan x = -\frac{3}{4}\).

Sine, Cosine, Tangent Relationships

Understanding the relationships among sine, cosine, and tangent is essential in trigonometry. These functions are interrelated and can often be derived from each other. They are usually described in terms of a right-angled triangle, though their values can be computed for any angle using their ratios and identities.

• \(\sin x\) is the opposite side over the hypotenuse.
• \(\cos x\) is the adjacent side over the hypotenuse.
• \(\tan x\) is the opposite side over the adjacent side, but it can also be expressed as \(\frac{\sin x}{\cos x}\).

Using these relationships, if you know one function, you can often deduce the others. In our solved example: - Given \(\sin x = \frac{3}{5}\), the Pythagorean Identity helped us find \(\cos x = -\frac{4}{5}\).- Using the definition \(\tan x = \frac{\sin x}{\cos x}\), we calculated \(\tan x = -\frac{3}{4}\).Understanding these relationships not only helps solve problems more efficiently but also deepens comprehension of how trigonometric functions operate as a system.