Problem 56 - Algebra 2

Question

Use half-angle identities to write each expression, using trigonometric functions of \(\theta\) instead of \(\frac{\theta}{4} .\) $$ \tan \frac{\theta}{4} $$

Step-by-Step Solution

Answer

The expression \( tan(\frac{\theta}{4}) \) can be written in terms of \( \theta \) as \( tan(\frac{\theta}{4}) = \pm \sqrt{\frac{1 - \sqrt{\frac{1 + cos \theta}{2}}}{1 + \sqrt{\frac{1 + cos \theta}{2}}}} \).

Step 1: Half-angle formulas derivation

Let's start by using the half-angle formula \( tan(\frac{y}{2}) = \pm \sqrt{\frac{1 - cos y}{1 + cos y}} \) where \( y = \frac{\theta}{2} \). This lets us resolve \( tan(\frac{\theta}{4}) \) into functions of \( \frac{\theta}{2} \).

Step 2: Evaluate the half-angle formula

By substituting \( y = \frac{\theta}{2} \) into the half-angle formula, \( tan(\frac{\theta}{4}) = \pm \sqrt{\frac{1 - cos(\frac{\theta}{2})}{1 + cos(\frac{\theta}{2})}} \).

Step 3: Apply half-angle formula again

Now, we want to further simplify the expression in terms of \( \theta \). We will use the half-angle formula for cosine, namely, \( cos \frac{x}{2} = \pm \sqrt{\frac{1 + cos x}{2}} \), where \( x = \theta \) to compute \( cos(\frac{\theta}{2}) \).

Step 4: Evaluate the half-angle formula for cosine

By substituting \( x = \theta \) into the half-angle formula for cosine, we get \( cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + cos \theta}{2}} \). Now, we can substitute this expression back into the ration from step 2.

Step 5: Final expression

By substituting \( cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + cos \theta}{2}} \) into the equation from Step 2, we find that \( tan(\frac{\theta}{4}) = \pm \sqrt{\frac{1 - \sqrt{\frac{1 + cos \theta}{2}}}{1 + \sqrt{\frac{1 + cos \theta}{2}}}} \). Thus, we've expressed \( tan(\frac{\theta}{4}) \) in terms of \( \theta \) as desired.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational in mathematics, especially in the study of triangles and circles. They relate angles to side lengths and are prevalent in various real-world applications including physics and engineering. The primary trigonometric functions are sine, cosine, and tangent. Each function has its own specific utility:

• Sine (\( \sin \theta \)) is used to compute the ratio of the length of the opposite side to the hypotenuse in a right triangle.
• Cosine (\( \cos \theta \)) represents the ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)) calculates the ratio of the opposite side to the adjacent side.

Understanding these functions helps in solving problems related to waves, oscillations, and various engineering tasks. Explore their relationships through identities and transformations, which allow angles to be split or combined, such as the half-angle identities. This makes complex calculations more manageable.

Tangent

The tangent function, or \( \tan \theta \), is significant when considering angles in trigonometry. It is the ratio of the length of the opposite side to the adjacent side in a right triangle. Unlike sine and cosine, tangent can be undefined, specifically when the angle \( \theta \) is an odd multiple of \( 90^ \circ \).Half-angle identities are a crucial tool. They allow the breakdown of an angle into half its measure to find a more straightforward form of the function:

• The formula for the tangent of half an angle is \( \tan(\frac{y}{2}) = \pm \sqrt{\frac{1 - \cos y}{1 + \cos y}} \), helping to derive expressions in terms of another variable, such as simplifying \( \tan(\frac{\theta}{4}) \).
• This transformation is valuable for simplifying or calculating angles that are not easily computed with standard trigonometric tables or tools.

Explore these identities to gain a deeper understanding of how trigonometric functions interact and transform.

Cosine

Cosine is one of the primary trigonometric functions representing the ratio of the length of the adjacent side to the hypotenuse in a right triangle. In our half-angle context, the cosine identity becomes invaluable:

• The half-angle formula for cosine is \( \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}} \).
• By employing this formula, you can express \( \cos(\frac{\theta}{2}) \), aiding in simplifying expressions like \( \tan(\frac{\theta}{4}) \).

These identities are essential for dealing with angles that are not directly accessible through basic calculations. Understanding the behavior of cosine and its transformations ensures better handling of complex problems involving rotations and oscillations. It allows us to express complex trigonometric expressions simply and comprehensively, aiding in both theoretical and practical applications.