Problem 9 - Precalculus Essentials

Question

Find the exact value of each expression. $$\cos ^{-1}\left(-\frac{\sqrt{2}}{2}\right)$$

Step-by-Step Solution

Answer

The exact value of \(\cos^{-1}\left(-\frac{\sqrt{2}}{2}\right)\) is \(3\pi/4\) in radian and \(135^{\circ}\) in degrees.

Step 1: Understand the Inverse Cosine

An inverse cosine function, denoted as \(\cos^{-1}(x)\) or \(\arccos(x)\), gives the angle (in radians or degrees) whose cosine is equal to \(x\). It gives the result in the interval \([0, \pi]\) or \([0^{\circ}, 180^{\circ}]\) for real values of \(x\). Therefore, we are looking for an angle in the previously mentioned interval whose cosine is \(-\sqrt{2}/2\).

Step 2: Recalling the Unit Circle

The cosine function gives the x-coordinate of a point on the unit circle corresponding to a given angle. The value \(-\sqrt{2}/2\) appears in the unit circle at angles \(3\pi/4\) and \(7\pi/4\). But in the principal range \([0, \pi]\) or \([0^{\circ}, 180^{\circ}]\) of \(\arccos(x)\), only the \(3\pi/4\) or \(135^{\circ}\) provides the right value for the cosine function.

Step 3: State the Solution

Therefore, the exact value of \(\cos^{-1}(-\sqrt{2}/2)\) is \(3\pi/4\) or \(135^{\circ}\).

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function often abbreviated as "cos." This function relates an angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. Essentially, for an angle \( \theta \) in a right triangle, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).

Cosine is particularly important because it extends beyond triangles to apply to functions and waves. This function oscillates between -1 and 1, creating a smooth wave pattern. The behavior of cosine repeats over a full circle every \(2\pi\) radians, or \(360^{\circ}\).

The inverse cosine, often called the arccosine and denoted as \( \cos^{-1} \) or \( \arccos \), is used to find the angle that corresponds to a particular cosine value. Its range is from \(0\) to \(\pi\) radians \([0^{\circ}, 180^{\circ}]\), providing us with the unique angle whose cosine is the specified number.

Unit Circle

The unit circle is a critical concept in trigonometry used to define trigonometric functions. It is a circle with a radius of one, centered at the origin of a coordinate plane. Every point \((x, y)\) on this circle satisfies the equation \(x^2 + y^2 = 1\).

The unit circle assists in understanding angles and trigonometric functions more comprehensively. Each angle on the unit circle corresponds to a specific point, where the x-coordinate represents the cosine of the angle, and the y-coordinate represents the sine.

For example, considering the point on the unit circle where the cosine is \(-\sqrt{2}/2\), the relevant angles are those that provide this x-value. In particular, \(3\pi/4\) radians (or \(135^{\circ}\)) lies within the specified range for the inverse cosine and gives us the correct value.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the included variables, relying only on trigonometric functions like sine, cosine, and tangent. These identities are instrumental in simplifying trigonometric expressions and solving equations.

• One of the most notable identities is the Pythagorean identity, \( \sin^2(\theta) + \cos^2(\theta) = 1 \), which is directly derived from the unit circle equation.
• The cosine of complementary angles identity, \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \), shows how sine and cosine are related through complementary angles.
• Cofunction identities, like \( \cos(\pi - \theta) = -\cos(\theta) \), reveal symmetries and periodicities inherent in trigonometric functions.

Using these identities can simplify solving inverse trigonometric functions, like determining the angle associated with a given cosine value, whether it is \(-\sqrt{2}/2\) as in our original problem.