Question

Use a double-angle identity for cosine to determine the half-angle formula for cosine, \(\cos \frac{x}{2}=\pm \sqrt{\frac{1+\cos x}{2}}\).

Short answer

\cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos(x)}{2}}.

Step-by-step solution

Understanding the double-angle identity

The double-angle identity for cosine is given by \[- \cos(2\theta) = 2(\cos^2 \theta) - 1\]. We will use this identity to derive the half-angle formula.

Express \(\cos(2\theta)\) in terms of \(\theta\)

Rewrite \[- \cos(2\theta) = 2(\cos^2 \theta) - 1\] to find \(\cos^2 \theta\): \[- 2\cos^2 \theta = \cos(2\theta) + 1 \].

Solve for \cos^2 \theta

Divide both sides by 2: \[- \cos^2 \theta = \frac{1 + \cos(2\theta)}{2}\].

Replace \(\theta\) with \(\frac{x}{2}\)

Let \( x = 2\theta \), so \( \theta = \frac{x}{2}\). Replacing \( \theta \) with \( \frac{x}{2} \), the equation becomes \[- \cos^2 \left( \frac{x}{2} \right) = \frac{1 + \cos(x)}{2}\].

Solve for \cos \left( \frac{x}{2} \right)

Take the square root of both sides to solve for \cos \left( \frac{x}{2} \right): \[- \cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos(x)}{2}}\].

Key concepts

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variable where both sides of the equation are defined. These identities help simplify complex trigonometric expressions and solve trigonometric equations. They are based on the fundamental properties of the trigonometric functions. Key trigonometric identities include:

• Pythagorean identities like \(\text{sin}^2\theta + \text{cos}^2\theta = 1\)
• Reciprocal identities such as \(\text{sec}\theta = \frac{1}{\text{cos}\theta}\)
• Quotient identities like \(\text{tan}\theta = \frac{\text{sin}\theta}{\text{cos}\theta}\)

These identities are fundamental when working with trigonometric problems, making them easier to manage.

Double-Angle Identity

The double-angle identity is a powerful trigonometric tool used to express trigonometric functions of double angles (2\theta) in terms of single angles (\theta). This identity is crucial in deriving other trigonometric formulas, such as half-angle formulas. For cosine, the double-angle identity is:

• \(\text{cos}(2\theta) = 2\text{cos}^2\theta - 1\)
• It can also be rearranged as \(\text{cos}(2\theta) = 1 - 2\text{sin}^2\theta\)

This identity helps express \(\text{cos}(2\theta)\) to facilitate solving equations and proving various trigonometric properties.

Cosine

Cosine (cos) is one of the primary trigonometric functions, representing the ratio of the adjacent side to the hypotenuse in a right triangle. Cosine values range from -1 to 1, oscillating periodically along the unit circle. Key properties of the cosine function include:

• It is an even function, meaning \(\text{cos}(-\theta) = \text{cos}(\theta)\)
• Its value at 0 degrees or 0 radians is 1, and at 90 degrees or \(\frac{\text{π}}{2}\) radians is 0
• It has a period of 360 degrees or 2π radians

These properties and the relationship with other trigonometric functions make cosine essential for solving trigonometric problems, including using the double-angle and half-angle formulas.