Problem 18 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

Simplify the expression. \(\cos \left(x-\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

\(-\sin(x)\

Step 1: Recognize the structure of the expression

The expression \(\cos \left(x-\frac{\pi}{2}\right)\) is a common pattern that should be recognized. The expression is of the form \(\cos(\pi/2 - x)\), where \(x\) is the variable.

Step 2: Apply cosine properties

The cosine function has a periodicity of \(2\pi\) and it's known that \(\cos(\pi/2 + x) = -\sin(x)\). In this case, the expression can be written as \(-\sin(x)\).

Step 3: Result

So, the simplified form of the expression \(\cos \left(x-\frac{\pi}{2}\right)\) is \(-\sin(x)\).

Key Concepts

Cosine Function Properties

Understanding the properties of the cosine function is crucial for simplifying trigonometric expressions. Essentially, the cosine function, written as \(\cos(x)\), is one of the primary trigonometric functions used to describe the relationship between the angles and sides of a right triangle. A fundamental property to remember is that cosine is an even function, implying that \(\cos(-x) = \cos(x)\). Also, the cosine function has a range from -1 to 1 and its graph is a wave that repeats every \(2\pi\) radians, which is known as its period.

The maximum value of \(\cos(x)\) is 1 when \(x\) is an integer multiple of \(2\pi\), and the minimum value of -1 is achieved when \(x\) is an odd integer multiple of \(\pi\). Moreover, we have specific values corresponding to angles that are commonly used in trigonometry, such as \(\cos(0) = 1\), and \(\cos(\frac{\pi}{2}) = 0\). Familiarity with these properties is a powerful tool when tackling trigonometric simplifications.

Trigonometric Periodicity

Trigonometric periodicity refers to the repeating nature of sine, cosine, and tangent functions. All trigonometric functions have periods, which is the length of the interval on the x-axis after which the function’s values start repeating. For the cosine and sine functions, the period is \(2\pi\) radians, meaning that \(\cos(x + 2\pi) = \cos(x)\) and \(\sin(x + 2\pi) = \sin(x)\).

Understanding periodicity allows one to transpose an angle within one period to simplify an expression. For instance, if a solution step mentions \(\cos(\frac{\pi}{2} + x) = -\sin(x)\), it uses both the periodicity and the phase shift property, suggesting that we can convert a cosine function into a sine function, while also potentially changing signs according to the quadrant of the angle. Periodicity is often used in conjunction with other trigonometric properties to reduce complex expressions to simpler, equivalent forms.

Trigonometric Identities

Trigonometric identities are equations that are true for all values of the variables involved and are extremely helpful in simplifying trigonometric expressions. Some of the most commonly used identities include the Pythagorean identity, \(\cos^2(x) + \sin^2(x) = 1\), the angle sum and difference identities, like \(\cos(x \pm y) = \cos(x)\cos(y) \mp \sin(x)\sin(y)\), and the double-angle identities, such as \(\cos(2x) = \cos^2(x) - \sin^2(x)\).

To simplify the expression \(\cos \left(x-\frac{\pi}{2}\right)\) with identities, we use the angle sum and difference identity, highlighting that \(\cos(\frac{\pi}{2} - x)\) is equal to \(\sin(x)\), but with a negative sign due to the quadrant where \(x-\frac{\pi}{2}\) lies. This is a perfect example of using identities to not only simplify an expression but also to demonstrate the inherently interconnected nature of trigonometric functions.