Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Use an identity to write each expression as a single trigonometric function value. $$\sqrt{\frac{1+\cos 76^{\circ}}{2}}$$

Step-by-Step Solution

Answer

The expression simplifies to \(\cos(38^{\circ})\).

Step 1: Recognize the Identity

The expression we have here, \(\sqrt{\frac{1+\cos 76^{\circ}}{2}}\), resembles the formula used for the cosine of half angles. Recall that \(\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1+\cos\theta}{2}}\). This is an established trigonometric identity.

Step 2: Determine Corresponding Angle

Given the expression is \(\sqrt{\frac{1+\cos 76^{\circ}}{2}}\), we identify it as \(\cos\left(\frac{76^{\circ}}{2}\right)\). Using the half-angle identity, set \(\theta = 76^{\circ}\), giving us the half-angle as \(\frac{76^{\circ}}{2} = 38^{\circ}\).

Step 3: Express Expression as a Single Trigonometric Function

By applying the half-angle identity, we can simplify \(\sqrt{\frac{1+\cos 76^{\circ}}{2}}\) to \(\cos(38^{\circ})\).

Step 4: Verify by Simplification

Verify that \(\cos(38^{\circ})\) correctly corresponds to the given identity, which is equivalent to the original expression when calculated or used in trigonometric tables.

Key Concepts

Half-Angle Identity

Understanding trigonometric identities can greatly simplify complex expressions. One such identity is the half-angle identity for cosine, which provides a method to simplify expressions involving square roots and cosine functions. The half-angle identity is represented as:\[\cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1+\cos\theta}{2}}\]This identity is particularly helpful when you encounter an expression like \(\sqrt{\frac{1+\cos 76^{\circ}}{2}}\). By recognizing the pattern that fits the identity, you can simplify it to a single trigonometric function value, as shown in the solution process.

• The main purpose of the half-angle identity is to convert complex expressions into more manageable trigonometric functions.
• It is especially useful for calculating angles not typically found in simple fraction degrees.

By applying this identity, you can simplify computations and ultimately verify your results against standard trigonometric values.

Cosine Function

Among the fundamental trigonometric functions, cosine measures the horizontal coordinate of a unit circle. It is prevalent in various mathematical applications, from geometry to calculus, due to its periodic nature. In the given expression, the cosine function plays a crucial role. By using the half-angle identity, \(\sin\) expressions transform into \(\cos\) expressions. This is particularly useful in simplifying square root expressions.

• Cosine is a periodic function with a period of 360° (or 2π radians).
• It is often used in scenarios requiring transformations of angled expressions into more straightforward calculations.

In the problem, the expression \(\sqrt{\frac{1+\cos 76^{\circ}}{2}}\) simplifies, through the cosine function's properties, to \(\cos(38^{\circ})\). Recognizing this simplification is vital to connecting trigonometric identities and real-world applications. The cosine function's ability to represent an angle in either the positive or negative direction also aids in ensuring that expressions remain simplified and true to their original representation. This reflects the importance of understanding the cosine function thoroughly in trigonometry.

Angle Simplification

Trigonometrical problem-solving often requires simplifying angles for easier computations. The concept of angle simplification allows for the transformation of less familiar angles into values more accessible and often found in trigonometric tables or calculators. In this exercise, simplifying the angle of 76° to 38° hinges on recognizing and applying the half-angle identity.

• Angle simplification reduces potentially confusing expressions to understandable and easily computible forms.
• It's crucial for calculations involving trigonometric tables, as exact angles are easier to locate and apply.

When using the identity, setting \(\theta = 76^{\circ}\) and realizing that this implies substituting the angle \(\frac{76^{\circ}}{2}\), or 38°, makes calculations clear and simple. This process is a common tactic in trigonometry, ensuring that more complex angles can still be handled with confidence. Embracing angle simplification not only aids in effective problem-solving but also strengthens the comprehension of trigonometric principles and relationships, making it a valuable skill in mathematics.