Question

Inverse sines and cosines Without using a calculator, evaluate the following expressions or state that the quantity is undefined. $$\cos ({\cos }^{-1}(-1))$$

Short answer

$$Answer:Thevalueoftheexpressionis$$-1$$.

Step-by-step solution

Identify the expression

We are given the expression: $$\cos ({\cos }^{-1}(-1))$$

Apply the properties of inverse trigonometric functions

Since the $\cos$ and ${\cos }^{-1}$ functions are inverses of each other, they will "cancel out" when applied in this order, provided that the argument of the inverse function is in the range of the inverse cosine function. The range of ${\cos }^{-1}$ is from $-1$ to $1$. Since $-1$ is within this range, we can proceed.

Evaluate the expression

Applying the cancellation property, we have: $$\cos ({\cos }^{-1}(-1))=-1$$ The expression evaluates to $-1$.

Key concepts

Cosine Function

The cosine function is one of the primary trigonometric functions used to describe the relationship between the sides of a right-angled triangle. Represented as $\text{cos}(x)$, where $x$ is an angle measured in radians, the function gives the ratio of the adjacent side to the hypotenuse of the triangle. For example, if we have a right-angled triangle where $x$ is one of the non-right angles, and 'a' and 'c' are the lengths of the adjacent side and the hypotenuse, respectively, the cosine of angle $x$ is $\text{cos}(x)=\frac{a}{c}$.

The cosine function has a range of [-1,1], meaning it will always return a value within this interval regardless of the input angle. This property is central to understanding why the inverse cosine (or arc cosine), denoted as ${\text{cos}}^{-1}(x)$, returned -1 when given -1 as the argument in the exercise solution.

Additionally, the cosine function is periodic with a period of $2\text{π}$. In other words, $\text{cos}(x)=\text{cos}(x+2\text{π}k)$, where $k$ is an integer. It's also even, meaning that $\text{cos}(-x)=\text{cos}(x)$, which can simplify the evaluation of expressions involving cosine.

Properties of Inverse Functions

Inverse functions essentially reverse the effect of the original function. If we have a function $f(x)$ that takes an input $x$ and gives an output $y$, then its inverse $f^{-1}(y)$ would take $y$ as the input and give us back $x$. For the inverse process to be well-defined, the original function must be one-to-one, meaning that each output is the result of one specific input.

In the realm of trigonometry, inverse trigonometric functions such as ${\text{cos}}^{-1}(x)$, also known as arc cosine, have a specific range to ensure they are one-to-one. For ${\text{cos}}^{-1}(x)$, this range is $[0,\text{π}]$ in radians.

An important aspect to note is that ${\text{cos}}^{-1}(x)$ and $\text{cos}(x)$ 'cancel' each other out, as seen in the solution. This means that $\text{cos}({\text{cos}}^{-1}(y))=y$, if $y$ is in the range of $\text{cos}$. The exercise improvement advice here would include noting that the cancellation property applies if and only if the value inside the inverse cosine is within its domain of [-1,1].

Trigonometric Identities

Trigonometric identities are equations that express one trigonometric function in terms of others, or establish a relationship between angle measures. These identities are invaluable for simplifying expressions and solving trigonometric equations.

One of the most fundamental identities is the Pythagorean identity: ${\text{cos}}^2(x)+{\text{sin}}^2(x)=1$, which is derived from the Pythagorean theorem relating to the sides of a right-angled triangle. There are also angle sum and difference identities, such as $\text{cos}(x\text{±}y)=\text{cos}(x)\text{cos}(y)\text{∓}\text{sin}(x)\text{sin}(y)$, which allow us to find the cosine of the sum or difference of two angles.

Using these identities, we can transform complex trigonometric equations into simpler forms that are more easily solvable. In the case of our exercise, we didn't have to employ these identities since the solution leaned heavily on understanding the property of inverse functions. However, in more complex problems, awareness of trigonometric identities, including those involving inverse functions, is crucial for efficient problem-solving.