Problem 37 - Precalculus Mathematics for Calculus

Question

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$\sin x=\frac{3}{3}, \quad 0^{\circ}

Step-by-Step Solution

Answer

\( \sin \frac{x}{2} = \frac{\sqrt{2}}{2}, \cos \frac{x}{2} = \frac{\sqrt{2}}{2}, \tan \frac{x}{2} = 1 \).

Step 1: Evaluate the given sine value

From the equation given, \( \sin x = \frac{3}{3} \) which simplifies to \( \sin x = 1 \). Since \( 0^{\circ} < x < 90^{\circ} \), this implies \( x = 90^{\circ} \).

Step 2: Determine \(x/2\)

Since \( x = 90^{\circ} \), calculate \( \frac{x}{2} = \frac{90^{\circ}}{2} = 45^{\circ} \).

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

Using the angle found, \( \sin \frac{x}{2} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \).

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

Similarly, \( \cos \frac{x}{2} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \).

Step 5: Calculate \( \tan \frac{x}{2} \)

Lastly, \( \tan \frac{x}{2} = \tan 45^{\circ} = 1 \).

Key Concepts

Half-Angle Formulas

The Half-Angle Formulas are a valuable tool in trigonometry, allowing us to find the trigonometric functions of angles that are not directly accessible through known identities. These formulas apply when you want to calculate the sine, cosine, or tangent of half of a given angle. They are especially useful in finding the trigonometric functions for angles like 22.5 degrees or 15 degrees.

Here are the half-angle formulas you should remember:

• The sine half-angle formula: \[\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}\] This formula helps you find the sine of half an angle when you know the cosine of the full angle.
• The cosine half-angle formula:\[\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \] Use this formula to find the cosine of half an angle if you know the cosine of the full angle.
• The tangent half-angle formula:\[\tan \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}}\]This equation assists in determining the tangent of half an angle using the cosine of the full angle.

In our scenario, since \(x=90^{\circ}\), these formulas simplify simply by using angles we already know, like 45 degrees.

Sine Function

The sine function is one of the primary trigonometric functions, represented as \( \sin x \). It denotes the ratio of the length of the opposite side to the hypotenuse in a right triangle. It's crucial for understanding wave patterns, oscillations, and the behavior of periodic functions.

For angles between 0 to 90 degrees, the sine function increases from 0 to 1. In our example, \( \sin x = 1 \), meaning that the angle \(x\) is 90 degrees. Understanding this could help you visualize the function's cyclical nature and its relationship within a unit circle.

• When \( \sin x = 1 \), the angle \(x\) being 90 degrees lies at the top of the unit circle, where the y-coordinate is exactly 1.
• If \( \sin x < 0 \), the angle is greater than 180 degrees within its cycle, below the x-axis.

With \(x=90^{\circ}\), half the angle or \(45^{\circ}\) gives \(\sin \frac{x}{2} = \frac{\sqrt{2}}{2}\). This ties back to the half angle formula. It shows the consistency and symmetry in trigonometry.

Cosine Function

The cosine function, denoted as \( \cos x \), represents another fundamental trigonometric function. It measures the ratio of the adjacent side to the hypotenuse in a right triangle. The cosine function is the counterpart of the sine function and plays an essential role in understanding rotations, periodic functions, and relationships in trigonometry.

As the angle moves from 0 to 90 degrees, the cosine value decreases from 1 to 0. Within this exercise, we initially find \(x\) equals 90 degrees, making the cosine \(0\) for \(x\). Then, with \(\frac{x}{2} = 45^{\circ}\), using the half-angle formula and known values,

• The cosine at this angle is \(\cos \frac{x}{2} = \cos 45^{\circ} = \frac{\sqrt{2}}{2}\).
• This value shows how both sine and cosine are equal at 45 degrees within their positive cycles.

This unity in values explains their essential roles in creating balanced periodic functions and symmetrical rotations on the unit circle.

Tangent Function

The tangent function, symbolized as \( \tan x \), is the ratio of the sine function to the cosine function, represented mathematically as \( \tan x = \frac{\sin x}{\cos x} \). The tangent function is beneficial for solving merit-based triangles in trigonometry and calculus.

In our problem, we are examining \( \tan \frac{x}{2} \), where \( x = 90^{\circ} \). Thus,

• \( \tan 45^{\circ} = 1 \), reflecting the tangent's unique property where the angle divides the plane equally.
• This is intuitive since \( \tan x = 1 \) implies equal sine and cosine values, often occurring where angles such as \(45^{\circ}\) or \(135^{\circ}\) lie.

This case solidifies the tangent's role in establishing the relationship between angles and their trigonometric functions effectively.