Problem 35 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\sin (\pi+x)$$

Step-by-Step Solution

Answer

\( \sin(\pi + x) = -\sin(x) \).

Step 1: Recognize the Angle Addition Identity

The given problem involves rewriting the expression \( \sin(\pi + x) \). Start by identifying the trigonometric identity applicable to the problem. The appropriate identity here is the angle addition identity for sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \).

Step 2: Apply the Identity to the Expression

Apply the angle addition identity to the expression \( \sin(\pi + x) \). Substitute \(a = \pi\) and \(b = x\) into the identity: \[ \sin(\pi + x) = \sin(\pi) \cos(x) + \cos(\pi) \sin(x). \]

Step 3: Evaluate Trigonometric Functions of \(\pi\)

Evaluate the trigonometric functions at \(\pi\). Recall from the unit circle that \( \sin(\pi) = 0 \) and \( \cos(\pi) = -1 \). Substitute these values into the equation from the previous step:\[ \sin(\pi + x) = 0 \cdot \cos(x) + (-1) \cdot \sin(x). \]

Step 4: Simplify the Expression

Simplify the expression by performing the multiplications:\[ \sin(\pi + x) = -\sin(x). \] Thus, the expression \( \sin(\pi + x) \) simplifies to \(-\sin(x)\).

Key Concepts

Angle Addition Formula

The angle addition formula is a fundamental tool in trigonometry that helps us find the sine, cosine, or tangent of the sum of two angles. Specifically for sine, the formula is expressed as:\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]This identity allows us to break down a complex trigonometric expression involving a sum of angles into simpler components. It involves the sine and cosine of each individual angle. So in our exercise, this formula helps us dissect \(\sin(\pi + x)\) by considering \(a = \pi\) and \(b = x\). Now, you can apply this identity seamlessly whenever faced with the sine of an added angle.

Sine Function

The sine function is one of the basic trigonometric functions. It relates to the y-coordinate of a point on the unit circle. For any angle \(\theta\), \(\sin \theta\) represents the vertical component of a point on the unit circle. In simpler terms,

• \(\sin\) is linked to the height at which the point on the circle is along the vertical axis.

The sine function is periodic with a period of \(2\pi\). This means it repeats its values in a regular pattern every \(2\pi\) interval. In our problem, understanding that \(\sin(\pi) = 0\) is key to simplifying our expression \(\sin(\pi + x)\).

Unit Circle

The unit circle is a crucial concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. Each point on this circle corresponds to an angle in radians and gives us the cosine and sine values for that angle.The circle allows us to easily visualize:

• \(\sin\) values as the y-coordinates.
• \(\cos\) values as the x-coordinates.

For example, knowing that at \(\pi\), the point on the unit circle is \((-1, 0)\), helps us immediately conclude \(\sin(\pi) = 0\) and \(\cos(\pi) = -1\). These forms directly influence the simplification process in our expression \(\sin(\pi + x)\).

Simplification

Simplification is often the final step in solving mathematical problems. It involves reducing an expression to its most straightforward form. Here, simplification means taking the expression \(\sin(\pi + x)\) and using known values from our angle addition formula application.After substituting \(\sin(\pi)\) and \(\cos(\pi)\) into the equation, we perform the following operations:

• \(0 \times \cos(x) = 0\)
• \(-1 \times \sin(x) = -\sin(x)\)

By carrying out these calculations, we simplify the complex trigonometric expression to just \(-\sin(x)\). This process illustrates the power of combining trigonometric identities and known values to make a solution more elegant and concise.