Problem 40 - Precalculus Mathematics for Calculus

Question

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$\tan x=1, \quad 0^{\circ}

Step-by-Step Solution

Answer

\( \sin \frac{x}{2} = \sqrt{\frac{2 - \sqrt{2}}{4}}, \cos \frac{x}{2} = \sqrt{\frac{2 + \sqrt{2}}{4}}, \tan \frac{x}{2} = \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}} \)

Step 1: Identify the Angle x

Given that \( \tan x = 1 \) and \( 0^{\circ} < x < 90^{\circ} \), we need to find the angle \( x \). From trigonometric identities, \( \tan 45^{\circ} = 1 \). Therefore, \( x = 45^{\circ} \).

Step 2: Calculate \( \sin \frac{x}{2} \)

Using the angle \( x = 45^{\circ} \), we want to find \( \sin \frac{x}{2} \). We use the identity: \[ \sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \]Since \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \), substitute: \[ \sin \frac{45^{\circ}}{2} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \]This simplifies to:\[ \sin 22.5^{\circ} = \sqrt{\frac{2 - \sqrt{2}}{4}} \]

Step 3: Calculate \( \cos \frac{x}{2} \)

Now, using the identity:\[ \cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \]Substitute \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \): \[ \cos \frac{45^{\circ}}{2} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \]This simplifies to:\[ \cos 22.5^{\circ} = \sqrt{\frac{2 + \sqrt{2}}{4}} \]

Step 4: Calculate \( \tan \frac{x}{2} \)

Using the identity:\[ \tan \frac{x}{2} = \sqrt{\frac{1 - \cos x}{1 + \cos x}} \]Substitute \( \cos 45^{\circ} = \frac{\sqrt{2}}{2} \): \[ \tan \frac{45^{\circ}}{2} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{1 + \frac{\sqrt{2}}{2}}} \]This simplifies to:\[ \tan 22.5^{\circ} = \sqrt{\frac{2 - \sqrt{2}}{2 + \sqrt{2}}} \]

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that relate the various trigonometric functions to one another. They're very useful in simplifying complex expressions and solving trigonometric equations. They include:

• Pythagorean identities: These relate the squares of the sine, cosine, and tangent functions, like \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Reciprocal identities: These express trigonometric functions as reciprocals, such as \( \csc \theta = \frac{1}{\sin \theta} \).
• Angle sum and difference identities: These indicate how to compute the sine, cosine, or tangent of angle sums or differences.

For this exercise, half-angle identities are particularly useful. They allow us to find the sine, cosine, and tangent of half an angle when you know the original angle. Specifically, half-angle identities are expressed as:

• \( \sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \)
• \( \cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \)
• \( \tan \frac{x}{2} = \sqrt{\frac{1 - \cos x}{1 + \cos x}} \)

These identities are grounded on the core trigonometric relationships and can simplify complex calculations significantly.

Angle Calculation

Angle calculation involves determining the value of an unknown angle based on given trigonometric values or identities. In our exercise, the known value is the tangent of the angle.

We start with \( \tan x = 1 \) and the condition that the angle is located in the first quadrant, \( 0^\circ < x < 90^\circ \). In trigonometry, it's common knowledge that \( \tan 45^\circ = 1 \), hence it's logical to conclude that \( x = 45^\circ \). This is due to the property of the tangent function, which indicates that it equals one when the angle is 45 degrees.
Additionally, it's crucial to ensure the angle is in the correct quadrant when solving trigonometric problems. Each trigonometric function has different significances in different quadrants.
Understanding these characteristics helps one expertly calculate unknown angles, which is foundational in broader geometry or physics problems.

Tangent Function

The tangent function, denoted as \( \tan \), is one of the primary trigonometric functions used to relate angles to side lengths in right-angled triangles. It's defined as the ratio:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

In the context of our exercise, we know \( \tan x = 1 \), which leads to the discovery of the angle being \( 45^\circ \). Remember, \( \tan 45^\circ = 1 \) because the sine and cosine of \( 45^\circ \) are equal, making their ratio 1.

Moreover, the tangent function plays a crucial role in the half-angle identity for tangent:

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

This identity allows us to find the tangent of half an angle based on the cosine of the original angle, which is particularly useful in many trigonometric applications.
Understanding the properties and applications of the tangent function can greatly assist in solving intricate trigonometric problems with efficiency and accuracy.