Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\sin \left(45^{\circ}+x\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{\sqrt{2}}{2} (\cos(x) + \sin(x)) \).

Step 1: Identify the Appropriate Trigonometric Identity

To solve the expression \( \sin(45^{\circ} + x) \), recognize that it can be rewritten using the sine angle addition identity: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Step 2: Substitute the Values into the Identity

Use the values \( a = 45^{\circ} \) and \( b = x \). Substitute into the identity: \( \sin(45^{\circ}) \cos(x) + \cos(45^{\circ}) \sin(x) \).

Step 3: Calculate Trigonometric Values for Known Angles

Calculate \( \sin(45^{\circ}) = \frac{\sqrt{2}}{2} \) and \( \cos(45^{\circ}) = \frac{\sqrt{2}}{2} \).

Step 4: Substitute Trigonometric Values into the Expression

Replace the calculated values into the expression: \( \frac{\sqrt{2}}{2} \cos(x) + \frac{\sqrt{2}}{2} \sin(x) \).

Step 5: Simplify the Expression

Factor out the common factor \( \frac{\sqrt{2}}{2} \) to write the expression as \( \frac{\sqrt{2}}{2} (\cos(x) + \sin(x)) \). This is the expression written as a function of \( x \) only.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in trigonometry, providing relationships between different trigonometric functions. They are equations true for all values of the included variables where both sides are defined. Trigonometric identities are used to simplify expressions and solve equations involving trigonometric functions.

Some common trigonometric identities include:

• Pythagorean Identities, such as \( \sin^2\theta + \cos^2\theta = 1 \).
• Reciprocal Identities, like \( \csc \theta = \frac{1}{\sin \theta} \).
• Tangent and Cotangent Identities, with \( \tan \theta = \frac{\sin \theta}{\cos \theta} \).

Understanding and using these identities can simplify complex problems and help in transforming expressions into more familiar or useful forms. In this exercise, we specifically make use of the Sine Angle Addition Identity. This identity allows us to expand and simplify the sine of a sum, transforming the expression into a more manageable function of \( x \).

Trigonometric Functions

Trigonometric functions are essential components of trigonometry, describing relationships between angles and sides of a triangle. The main functions include sine, cosine, and tangent, often abbreviated as \( \sin \), \( \cos \), and \( \tan \), respectively. These functions are fundamental in various applications ranging from geometry to real-world modeling.

Here are what these functions represent:

• \( \sin \theta \): The ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
• \( \cos \theta \): The ratio of the length of the adjacent side to the hypotenuse.
• \( \tan \theta \): The ratio of the sine to the cosine of the angle, equivalent to the opposite over adjacent.

These functions are periodic and repeat their values in regular intervals. For example, sine and cosine functions have a period of \( 2\pi \), while the tangent function has a period of \( \pi \). In trigonometric equations, identifying and utilizing the properties of these functions, such as periodicity and symmetry, can greatly aid in solving and simplifying problems as seen in the original exercise.

Angle Addition Formulas

The angle addition formulas are critical tools when dealing with expressions involving sums of angles. These formulas allow you to break down complex trigonometric expressions into simpler parts. Specifically, the sine and cosine angle addition formulas are quite useful. The formulas are as follows:

• For sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).
• For cosine: \( \cos(a + b) = \cos a \cos b - \sin a \sin b \).

These identities are used when angles are not standard angles, like 30°, 45°, 60°, or multiples thereof.

In the problem, we used the sine angle addition formula to simplify \( \sin(45^{\circ}+x) \). By breaking down the original expression into known trigonometric values and expressions only involving \( x \), we expressed the original problem in terms of a simpler function. This method not only simplifies the expression but also reveals deeper insights into the relationships between trigonometric functions and their arguments.