Question

Draw a right triangle to simplify the given expressions. $$\tan \left(\cos ^{-1} x\right)$$

Short answer

In the given expression, we have the inverse cosine function and the tangent function. By creating a right triangle, we can represent the expression using the properties of the triangle. We draw a right triangle ABC, with angle A being the target angle, and find the lengths of sides BC and AB using the cosine function and the Pythagorean theorem, respectively. The final simplified expression for $\tan \left(\cos ^{-1} x\right)$ is found to be: $$\tan \left(\cos ^{-1} x\right) = \frac{\sqrt{1 - x^2}}{x}$$

Step-by-step solution

Understand the expression

The given expression involves two trigonometric functions: the inverse cosine function, written as cos^(-1)(x), and the tangent function, written as tan(). We want to find the tangent of the angle whose cosine is x. Since we need to find the tangent of an angle, we should draw a right triangle with an angle whose cosine is x.

Draw a right triangle

Let's consider a right triangle ABC, with right angle at B, angle A as our target angle, and angle C as the remaining angle. We know from the problem that: $$\cos A = x$$ By definition, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse. Let's assume BC is the adjacent side and AC is the hypotenuse. So, $$\cos A = \frac{BC}{AC} = x$$ That means BC = x * AC.

Finding the third side

Now, we need to determine the length of the remaining side AB. We can do this using the Pythagorean theorem, which states that in a right triangle: $$\text{hypotenuse}^2 = \text{adjacent side}^2 + \text{opposite side}^2$$ In our triangle, we get: $$AC^2 = BC^2 + AB^2$$ Substitute BC with x * AC: $$AC^2 = (x * AC)^2 + AB^2$$ Now, divide by AC^2: $$1 = x^2 + \frac{AB^2}{AC^2}$$ Therefore, we find the ratio of the opposite side to the hypotenuse: $$\frac{AB}{AC} = \sqrt{1 - x^2}$$

Find the expression for the tangent

Finally, we can find the tangent of angle A, which is the ratio of the opposite side (AB) to the adjacent side (BC). Using the ratios we found in the previous steps, we get: $$\tan A = \frac{AB/AC}{BC/AC} = \frac{\frac{AB}{AC}}{x} = \frac{\sqrt{1 - x^2}}{x}$$ So, our final simplified expression is: $$\tan \left(\cos ^{-1} x\right) = \frac{\sqrt{1 - x^2}}{x}$$

Key concepts

Right Triangle

A right triangle is a type of triangle that contains a right angle, which is an angle of exactly 90 degrees. In geometry, right triangles are significant because they play a fundamental role in trigonometry. They help us understand relationships between the sides and angles of triangles. When dealing with a right triangle, one angle is typically labeled as 90 degrees, leaving the other two to be complementary, meaning they add up to 90 degrees together.

When working on problems involving trigonometric functions, it's common to draw a right triangle to visualize the relationship between the sides and the angles. A right triangle consists of three sides:

• The hypotenuse, which is the longest side opposite the right angle.
• The adjacent side, which is next to the angle of interest.
• The opposite side, which is directly across from the angle of interest.

By labeling these sides in context with an angle, like angle A in our exercise, it becomes much easier to apply trigonometric ratios to solve problems.

Pythagorean Theorem

The Pythagorean Theorem is a cornerstone of geometry, particularly when it comes to right triangles. This theorem states that in a right triangle, the square of the hypotenuse's length is equal to the sum of the squares of the other two sides. This relationship can be expressed in the equation:\[ c^2 = a^2 + b^2 \]where \( c \) represents the hypotenuse and \( a \) and \( b \) are the other two sides.

In the context of our problem, we're tasked with finding the third side of a triangle given some specific conditions. This is where the Pythagorean Theorem comes in handy. We use it to solve for the length of the side opposite angle A, once the lengths or relationships of the adjacent side and the hypotenuse are known. If the hypotenuse is denoted by \( AC \) and the adjacent side (to angle A) by \( BC \), the length of the opposite side (\( AB \)) can be calculated as:\[ AB = \sqrt{AC^2 - BC^2} \]This simple yet powerful theorem allows us to find unknown side lengths in right triangles, which can then be used to calculate trigonometric functions.

Tangent Function

The tangent function is a fundamental trigonometric function that helps relate the angles of a right triangle to its sides. For a given angle in a right triangle, the tangent function is defined as the ratio of the length of the side opposite to the angle over the length of the side adjacent to the angle. The mathematical expression for the tangent function is given by:\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

In our problem, we are interested in finding \( \tan(A) \), where \( A \) is the angle whose cosine is given by \( x \). This involves taking the opposite side \( AB \) over the adjacent side \( BC \). From the Pythagorean relationship and the setup given, we found:\[ \tan A = \frac{\sqrt{1 - x^2}}{x} \]The tangent function is particularly useful for calculating steepness or slopes, which is evident in other fields such as physics and engineering.

Trigonometric Ratios

Trigonometric ratios are vital in connecting angles and lengths in right triangles, and they form the basis of trigonometry. There are three primary trigonometric ratios: sine, cosine, and tangent. These ratios help to explore the properties of angles and are defined as follows:

• Sine (\( \sin \theta \)): the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \theta \)): the ratio of the adjacent side to the hypotenuse.
• Tangent (\( \tan \theta \)): the ratio of the opposite side to the adjacent side.

In the exercise at hand, we've used the cosine and tangent ratios. The cosine ratio helped us identify the relationship between the adjacent side and the hypotenuse. We drew from this to subsequently find the tangent ratio, essential for solving \( \tan(\cos^{-1}(x)) \). Understanding these ratios is crucial as they enable us to translate geometric intuition into mathematical expressions that can be worked on algebraically, greatly enhancing our problem-solving toolkit.