Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Use a half-number identity to find an expression for the exact value for each function, given the information about \(x\). $$\tan \frac{x}{2}, \text { given } \sin x=\frac{3}{5} \text { and } \frac{\pi}{2} < x < \pi$$

Step-by-Step Solution

Answer

\(\tan \frac{x}{2} = 3\)

Step 1: Identify the Double Angle Identity

We are given \(\sin x = \frac{3}{5}\). Since \( \frac{\pi}{2} < x < \pi \), \(x\) is in the second quadrant, and therefore \(\cos x < 0\). We need to use the identity \( \tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} \) to find \( \tan \frac{x}{2} \).

Step 2: Find \(\cos x\) Using Pythagorean Identity

We know \(\sin^2 x + \cos^2 x = 1\). Substituting \(\sin x = \frac{3}{5}\), we have \(\left(\frac{3}{5}\right)^2 + \cos^2 x = 1\). This simplifies to \(\frac{9}{25} + \cos^2 x = 1\), so \(\cos^2 x = \frac{16}{25}\). Therefore, \(\cos x = -\frac{4}{5}\) since \(x\) is in the second quadrant.

Step 3: Substitute \(\cos x\) Into the Half-Angle Identity

Substitute \(\cos x = -\frac{4}{5}\) into the identity \(\tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} \). This gives us \(\tan \frac{x}{2} = \pm \sqrt{\frac{1 + \frac{4}{5}}{1 - \frac{4}{5}}} = \pm \sqrt{\frac{\frac{9}{5}}{\frac{1}{5}}} = \pm \sqrt{9} = \pm 3\).

Step 4: Determine the Sign of \(\tan \frac{x}{2}\)

Since \(x\) is in the second quadrant, \(\frac{x}{2}\) is in the first quadrant. Thus, \(\tan \frac{x}{2}\) is positive. Therefore, \(\tan \frac{x}{2} = 3\).

Key Concepts

Half-Angle Identities

Half-angle identities are essential tools in trigonometry. They help us find values of trigonometric functions involving half angles. These identities are particularly useful when the angle is challenging to work with directly. In this exercise, we're focusing on the half-angle identity for tangent:

• \( an \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\)

The identity involves both sine and cosine functions. You use it to calculate the tangent of half of a given angle. The "+" or "-" sign depends on the quadrant in which the angle \/2 resides. In our example, since \(x\) is in the second quadrant, \(\frac{x}{2}\) is in the first quadrant where tangent is positive. Therefore, we select the positive option for \(\tan \frac{x}{2}\), resulting in \(\tan \frac{x}{2} = 3\). This principle can be broadly applied in solving various trigonometric problems.

Trigonometric Functions

Trigonometric functions methodically represent relationships between the angles and sides of a triangle. In this exercise, our primary focus is on sine, cosine, and tangent.

• Sine (\(\sin\)): It represents the ratio of the length of the side opposite to the angle to the hypotenuse.
• Cosine (\(\cos\)): It signifies the ratio between the length of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): It expresses the ratio of \(\sin x\) to \(\cos x\), or the opposite side to the adjacent side.

In our exercise, given \(\sin x = \frac{3}{5}\), and knowing that \(x\) is in the second quadrant, it helps us evaluate \(\cos x\) using the Pythagorean identity. Knowing these identities will make completing the half-angle identity task much simpler and quicker.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry. It states:

• \(\sin^2 x + \cos^2 x = 1\)

This identity holds for all angles and expresses the intrinsic relationship between sine and cosine functions. We use this identity to find unknown trigonometric values when others are given. For instance, if \(\sin x\) is known, you can seamlessly solve for \(\cos x\), or vice versa. In our solution, we use the Pythagorean identity to solve for \(\cos x\). Given \(\sin x = \frac{3}{5}\), we substitute into the identity:

• \(\left(\frac{3}{5}\right)^2 + \cos^2 x = 1\)

This allows us to find \(\cos^2 x = \frac{16}{25}\), leading to \(\cos x = -\frac{4}{5}\) since \(x\) is in the second quadrant. Understanding how to use this identity makes solving trigonometric problems much easier.