In Exercises 83–94, use a right triangle to write each expression as an algebraic expression. Assume that x is positive and that the given inverse trigonometric function is defined for the expression in x. tan (cos⁻¹ x)

Verified step by step guidance

1. <insert step 1> Start by understanding that ${\cos }^{-1}(x)$ represents an angle $\theta$ such that $\cos (\theta )=x$.
2. <insert step 2> Consider a right triangle where $\theta$ is one of the angles. Since $\cos (\theta )=\frac{\text{adjacent}}{\text{hypotenuse}}$, set the adjacent side to $x$ and the hypotenuse to 1.
3. <insert step 3> Use the Pythagorean theorem to find the opposite side: $\text{opposite}=\sqrt{1^2-x^2}=\sqrt{1-x^2}$.
4. <insert step 4> Now, find $\tan (\theta )$ which is $\frac{\text{opposite}}{\text{adjacent}}=\frac{\sqrt{1-x^2}}{x}$.
5. <insert step 5> Therefore, $\tan ({\cos }^{-1}(x))$ can be expressed as $\frac{\sqrt{1-x^2}}{x}$.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as cos⁻¹ (arccosine), are used to find angles when the value of a trigonometric function is known. For example, if cos(θ) = x, then θ = cos⁻¹(x). Understanding how to interpret these functions is crucial for solving problems involving angles and their relationships to side lengths in right triangles.

Right Triangle Relationships

In a right triangle, the relationships between the angles and sides are defined by trigonometric ratios: sine, cosine, and tangent. For instance, if we know an angle θ, the tangent of θ is the ratio of the opposite side to the adjacent side. This concept is essential for converting inverse trigonometric functions into algebraic expressions involving the sides of a triangle.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. Key identities, such as tan(θ) = sin(θ)/cos(θ), allow us to manipulate and simplify expressions involving trigonometric functions. Recognizing and applying these identities is vital for transforming expressions like tan(cos⁻¹(x)) into algebraic forms.