Problem 42 - Thomas Calculus

Question

In Exercises \(39-42,\) express the given quantity in terms of \(\sin x\) and \(\cos x\). $$\cos \left(\frac{3 \pi}{2}+x\right)$$

Step-by-Step Solution

Answer

The expression \( \cos \left(\frac{3\pi}{2} + x\right) \) is \( \sin(x) \).

Step 1: Understand the Trigonometric Identity

First, recall the trigonometric identity for cosine of a sum angle: \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \). We will use this identity to express \( \cos \left(\frac{3\pi}{2} + x\right) \) in terms of \( \sin x \) and \( \cos x \).

Step 2: Apply the Identity to the Given Expression

Apply the identity \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \) where \( a = \frac{3\pi}{2} \) and \( b = x \). So, \( \cos \left(\frac{3\pi}{2} + x\right) = \cos \left(\frac{3\pi}{2}\right)\cos(x) - \sin \left(\frac{3\pi}{2}\right)\sin(x) \).

Step 3: Evaluate the Cosine and Sine of Constants

Evaluate \( \cos \left(\frac{3\pi}{2}\right) \) and \( \sin \left(\frac{3\pi}{2}\right) \). Since \( \frac{3\pi}{2} \) is an angle on the negative y-axis, we know: \( \cos \left(\frac{3\pi}{2}\right) = 0 \) and \( \sin \left(\frac{3\pi}{2}\right) = -1 \).

Step 4: Substitute the Values into the Expression

Substitute the evaluated values into the expression: \( \cos \left(\frac{3\pi}{2} + x\right) = 0 \cdot \cos(x) - (-1) \cdot \sin(x) \). Simplifying, we have \( \cos \left(\frac{3\pi}{2} + x\right) = \sin(x) \).

Step 5: Final Expression

Thus, the expression \( \cos \left(\frac{3\pi}{2} + x\right) \) can be expressed as \( \sin(x) \), using the sine and cosine identities.

Key Concepts

Cosine of Sum Angle

The cosine of a sum angle is a fundamental trigonometric identity that helps you find the cosine of the sum of two angles. This identity is expressed as \( \cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b) \). By using this formula, one can easily break down complex angles into simpler components, making calculations more manageable.

For example, when working with \( \cos\left(\frac{3\pi}{2} + x\right) \), we identify \( a = \frac{3\pi}{2} \) and \( b = x \). Substituting these into our formula, we get \( \cos\left(\frac{3\pi}{2}\right)\cos(x) - \sin\left(\frac{3\pi}{2}\right)\sin(x) \). Breaking down angles this way not only simplifies computations but also ties together different trigonometric concepts.

The power of this identity lies in its ability to help with transformations and evaluations, essential in both simple and intricate trigonometric problems. It shows how different angle measures interact through sine and cosine values.

Sine and Cosine Evaluation

Evaluating sine and cosine values is a crucial step when working with trigonometric identities, especially when those involve specific angle measures. Calculating \( \cos\left(\frac{3\pi}{2}\right) \) and \( \sin\left(\frac{3\pi}{2}\right) \) involves understanding their positions on the unit circle.

- For \( \frac{3\pi}{2} \), this angle is located on the negative y-axis.
- Here, \( \cos\left(\frac{3\pi}{2}\right) \) is 0 because the point is directly on the vertical axis.
- \( \sin\left(\frac{3\pi}{2}\right) \) is -1 since it points downwards along the y-axis.
Knowing these specific trigonometric values simplifies the process of transforming expressions. For instance, substituting into the identity \( \cos\left(\frac{3\pi}{2} + x\right) = \cos\left(\frac{3\pi}{2}\right)\cos(x) - \sin\left(\frac{3\pi}{2}\right)\sin(x) \) gives \( 0\cdot\cos(x) - (-1)\cdot\sin(x) = \sin(x) \). This evaluation highlights how these core values are foundational to solving trigonometric problems efficiently.

Angle Transformation

Angle transformation in trigonometry is the process of converting expressions into simpler forms, making them easier to interpret and calculate. By transforming \( \cos\left(\frac{3\pi}{2} + x\right) \) into \( \sin(x) \), we employ the principles of trigonometric identities to arrive at a straightforward solution.

Transformation helps in:

• Converting complex sum angles into simple sine or cosine expressions.
• Simplifying the process of solving trigonometric equations.
• Developing deeper insights into how angles and their trigonometric values relate to one another.

These transformations are particularly useful in calculus, physics, and engineering, where trigonometric equations frequently appear.

Understanding and practicing transformations of angles is a valuable skill. It encourages a more profound appreciation of how trigonometric identities function and their wide applicability in solving real-world problems. By practicing angle transformations, one can enhance both their mathematical fluency and problem-solving abilities.