Problem 33 - A Graphical Approach to Precalculus with Limits

Question

Use identities to write each expression as a function with \(x\) as the only argument. $$\cos \left(\frac{3 \pi}{2}+x\right)$$

Step-by-Step Solution

Answer

The expression simplifies to \(\sin(x)\).

Step 1: Identify the Trigonometric Identity

The expression \(\cos \left(\frac{3 \pi}{2}+x\right)\) can be simplified using the cosine addition identity. This identity is: \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\).

Step 2: Assign Values to the Identity

In the expression \(\cos \left(\frac{3 \pi}{2}+x\right)\), assign \(a = \frac{3\pi}{2}\) and \(b = x\). Now apply the identity: \[\cos \left(\frac{3 \pi}{2} + x\right) = \cos \left(\frac{3 \pi}{2}\right) \cos(x) - \sin \left(\frac{3 \pi}{2}\right) \sin(x)\].

Step 3: Compute Cosine and Sine Values

The values of cosine and sine for \(\frac{3 \pi}{2}\) radians are: \(\cos \left(\frac{3 \pi}{2}\right) = 0\) and \(\sin \left(\frac{3 \pi}{2}\right) = -1\). Substitute these values into the equation: \[0 \cdot \cos(x) - (-1) \cdot \sin(x)\].

Step 4: Simplify the Expression

Simplify the expression: \[0 \cdot \cos(x) + \sin(x) = \sin(x)\]. This shows that the given expression simplifies to \(\sin(x)\).

Key Concepts

Cosine Addition Identity

The cosine addition identity is a fundamental concept in trigonometry that helps simplify expressions involving the sum of angles. It provides a formula to express the cosine of the sum of two angles, say \(a\) and \(b\), as a combination of sines and cosines of those angles. The identity is written as:

• \(\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)\)

This identity is particularly useful when you encounter expressions such as \(\cos\left(\frac{3\pi}{2} + x\right)\) because it allows you to break down more complicated functions into simpler trigonometric terms.
By assigning \(a = \frac{3\pi}{2}\) and \(b = x\), you can transform the original expression into one involving simply cosine and sine functions of known angles.
Computing these angles gives you specific values that can be substituted back into the expression to make it easier to handle. This method reduces the complexity of the trigonometric function to a more manageable form which, in some cases, is crucial for solving problems in calculus and physics.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using identities to rewrite them in alternative forms. This process can turn a seemingly complex expression into something simple and elegant. For the expression \(\cos\left(\frac{3\pi}{2} + x\right)\), using the cosine addition identity allows us to rewrite it as \(\cos\left(\frac{3\pi}{2}\right)\cos(x) - \sin\left(\frac{3\pi}{2}\right)\sin(x)\).

• Knowing that \(\cos\left(\frac{3\pi}{2}\right) = 0\) and \(\sin\left(\frac{3\pi}{2}\right) = -1\), the expression simplifies to \(0 \cdot \cos(x) + \sin(x)\).

Understanding the values for cosine and sine at key angles like \(\frac{3\pi}{2}\) is crucial here. It leads to further simplification:

• \(0 \cdot \cos(x) + \sin(x) = \sin(x)\)

This final simplification can make many types of trigonometric analysis much easier, whether you are evaluating an integral, solving an equation, or analyzing waves.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians provide a natural measure since they are based on the circle's radius.One full circle is \(2\pi\) radians, which matches the circumference formula of a circle \(C = 2\pi r\), where \(r\) is the radius.

• When we say \(\frac{3\pi}{2}\) radians, we mean an angle of 270 degrees, which is three-quarters of the way around the circle.
• Such understanding allows us to find the cosine and sine of angles, as we know exactly where they fall on the unit circle. At \(\frac{3\pi}{2}\) radians, the coordinates on the unit circle are \((0, -1)\), meaning \(\cos\left(\frac{3\pi}{2}\right) = 0\) and \(\sin\left(\frac{3\pi}{2}\right) = -1\).

Choosing to work in radians is advantageous in calculus and higher-level mathematics because many formulas, such as derivatives and integrals of trigonometric functions, assume angle measurements in radians for more natural and seamless calculations.