Problem 32 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\cos \left(\cos ^{-1} \frac{\sqrt{2}}{2}-\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

The value is 0.71.

Step 1: Understand \\( ext{cos}^{-1}\\)

\(\text{cos}^{-1}(x)\) refers to the angle \(\theta\) such that \(\cos(\theta) = x\). Given \(\text{cos}^{-1}\left(\frac{\sqrt{2}}{2}\right)\), we know \(\theta\) is the angle where \(\cos(\theta) = \frac{\sqrt{2}}{2}\). This corresponds to \(\theta = \frac{\pi}{4}\) because the cosine of \(\frac{\pi}{4}\) is \(\frac{\sqrt{2}}{2}\).

Step 2: Substitute the angle into the expression

Substitute \(\text{cos}^{-1}\left(\frac{\sqrt{2}}{2}\right)\) with \(\frac{\pi}{4}\) in the given expression: \(\cos\left(\frac{\pi}{4} - \frac{\pi}{2}\right)\).

Step 3: Simplify the angle within the cosine function

Subtract \(\frac{\pi}{2}\) from \(\frac{\pi}{4}\) to simplify the expression to \(\cos\left(\frac{\pi}{4} - \frac{\pi}{2}\right) = \cos\left(-\frac{\pi}{4}\right)\).

Step 4: Calculate cosine of the simplified angle

The cosine function is even, meaning \(\cos(-x) = \cos(x)\). Thus, \(\cos\left(-\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right)\). Since we know \(\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\), the result is \(\frac{\sqrt{2}}{2}\).

Step 5: Round the result to the nearest hundredth

Convert \(\frac{\sqrt{2}}{2}\) to a decimal: approximately 0.707. Rounded to the nearest hundredth, this value is 0.71.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are quite fascinating because they allow us to find angles when given specific trigonometric values. Here, we focus on the inverse cosine function, denoted as \( ext{cos}^{-1}(x)\). This function answers the question: "For which angle \(\theta\) is the cosine value equal to \(x\)?" If you have a number like \(\frac{\sqrt{2}}{2}\), using \(\text{cos}^{-1}\), you can discover that the corresponding angle is \(\frac{\pi}{4}\) radians. This angle makes sense because the cosine of \(\frac{\pi}{4}\) yields exactly \(\frac{\sqrt{2}}{2}\).

• Remember, each inverse trigonometric function has a specific range. For \(\text{cos}^{-1}\), the range is from 0 to \(\pi\) radians.
• This means when you solve \(\text{cos}^{-1}(x)\), the angle is always between 0 and \(\pi\) radians.

This kind of knowledge helps you solve trigonometric problems by flipping the usual process: instead of finding a trigonometric value from an angle, we find an angle from a trigonometric value.

Angle Measures in Radians

When dealing with trigonometry, it is crucial to understand angle measures in radians. Radians offer a more natural way of expressing angles as part of the circle's geometry. To visualize this, a full circle measures \(2\pi\) radians, similar to how it is 360 degrees. This comes from the circumference of a circle formula \((2\pi r)\), where the radius \(r\) is set to 1.

• One radian is the angle formed when the arc length is equal to the circle's radius.
• This relationship simplifies many trigonometric computations, making radians especially useful in calculus and beyond.

For example, \(\frac{\pi}{4}\) radians is \(45^\circ\) because it's a quarter of \(\pi\) (half-circle), and taking \(\frac{\pi}{2}\) radians equals \(90^\circ\). In calculations, converting between degrees and radians, you use the relation \(\pi\) rad = 180°.

Cosine Function

Understanding the cosine function is essential in trigonometry. The cosine of an angle gives the ratio of the adjacent side to the hypotenuse in a right triangle. The function itself is periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) radians. A key property of the cosine function is that it is even. This indicates that \(\cos(-x) = \cos(x)\).

• This property is useful for simplifying expressions, such as turning \(\cos(-\frac{\pi}{4})\) into \(\cos(\frac{\pi}{4})\).
• The cosine function ranges between -1 and 1, offering values like \(\frac{\sqrt{2}}{2}\) at specific angles like \(\frac{\pi}{4}\) radians.

In the given exercise, finding \(\cos(\frac{\pi}{4} - \frac{\pi}{2}) = \cos(-\frac{\pi}{4})\), and using the even function property, we confirm the value is \(\frac{\sqrt{2}}{2}\). It is rounded to 0.71 when expressed in decimal form and taken to the hundredth place. Grasping these fundamental properties enables you to tackle more complex trigonometric problems with ease.