Question

Right-triangle relationships Use a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\sec ^{-1} x\right)$$

Short answer

Answer: \(\frac{1}{x}\).

Step-by-step solution

Recall the relationship between secant and cosine

The secant function is the reciprocal of the cosine function. This means that: $$\sec(\theta) = \frac{1}{\cos(\theta)}$$ Now, given that we want to simplify the expression \(\cos(\sec^{-1}(x))\), we can apply the inverse function rule: $$x = \sec(\theta) \Rightarrow \theta = \sec^{-1}(x)$$

Define a right triangle

Let's define a right triangle with an angle \(\theta\), such that the secant of the angle \(\theta\) is equal to \(x\). To do this, let the hypotenuse be of length \(x\) and the adjacent side be of length 1. Then, the third side (opposite to the angle \(\theta\)) will be of length \(\sqrt{x^2 - 1}\) (according to the Pythagorean theorem). It is important to note that since \(x>0\), the angle \(\theta\) is acute (0 < \(\theta\) < \(\dfrac{\pi}{2}\)).

Apply the relationship between secant and cosine

Now, we have the right triangle with the angle \(\theta\), hypotenuse of length \(x\), adjacent side of length 1, and opposite side of length \(\sqrt{x^2 - 1}\). Recall that the cosine function is given by: $$\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}$$

Find the cosine of the angle

Now, using the triangle we defined in step 2, we can find the cosine of the angle as follows: $$\cos(\theta) = \frac{1}{x}$$ Since \(\theta = \sec^{-1}(x)\), we have: $$\cos(\sec^{-1}(x)) = \frac{1}{x}$$ So, the simplified expression is: $$\cos(\sec^{-1}(x)) = \frac{1}{x}$$

Key concepts

Secant Function

The secant function is a trigonometric function that represents the reciprocal of the cosine function. It is not as commonly used as its more familiar counterparts, sine and cosine, but it plays an essential role in trigonometry. In a right-angled triangle, if one angle is \( \theta \) and the length of the side adjacent to \( \theta \) is denoted as \( a \), and the hypotenuse is denoted as \( h \), the secant of \( \theta \) is defined as \( \sec(\theta) = \frac{h}{a} \).
In the context of the example exercise, the secant function is applied inversely, which means if the secant of an angle is given as \( x \), we can find the angle itself by calculating \( \sec^{-1}(x) \). This process involves defining a right-angled triangle with sides that correspond to the value of \( x \) as the hypotenuse and constructing the triangle accordingly.

Cosine Function

The cosine function is one of the primary trigonometric functions and relates the angle of a right-angled triangle to the ratio of the length of the adjacent side over the hypotenuse. Mathematically, it is expressed as \( \cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).
When given an angle \( \theta \), you can find the cosine of that angle by dividing the length of the side next to the angle ( not the hypotenuse) by the length of the hypotenuse. In practical scenarios, like the given exercise, the cosine function helps in finding the ratio relevant to the angle, especially after using inverse trigonometric functions to determine the angle itself. Understanding the interplay between the secant and cosine functions is crucial, as this exercise demonstrates, since \( \sec(\theta) \) is simply the reciprocal of \( \cos(\theta) \).

Inverse Trigonometric Functions

Inverse trigonometric functions are the reverse operations of the standard trigonometric functions. They are used to determine an angle when a ratio of sides in a right triangle is known. For instance, the inverse cosine function, denoted as \( \cos^{-1} \), finds an angle when the cosine ratio is given.
Just like the inverse cosine function, the inverse secant function, denoted as \( \sec^{-1} \), finds the angle when the secant value is known. In the given exercise, the challenge is to work backwards from the secant value \( x \) to find the cosine of the angle \( \theta \). Understanding these inverse functions is essential for solving trigonometric equations and simplifying expressions involving trigonometric identities.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be written as \( a^2 + b^2 = c^2 \), where \( c \) is the length of the hypotenuse, and \( a \) and \( b \) are the lengths of the other two sides.
In the context of trigonometric problems, the Pythagorean theorem helps to deduce one side of a triangle when the other two are known. This understanding is crucial in simplifying trigonometric expressions where, for example, the value of a secant function leads to determining the length of the sides of a triangle to find the cosine value, just as it was used in the step-by-step solution.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions entails using trigonometric identities and relationships to write the expressions in a more straightforward or more useful form. This often includes utilizing functions' properties, like the reciprocal nature of secant and cosine, to convert between different trigonometric functions.
Simplification can also involve applying the Pythagorean theorem to relate the sides of a right triangle, or using inverse trigonometric functions to find the angles from known ratios. The importance of simplification comes to light when solving equations or making calculations, as seen in the exercise, where using these relationships and principles produced the simplified expression \( \cos(\sec^{-1}(x)) = \frac{1}{x} \). The process not only offers a solution but also deepens comprehension of how these trigonometric concepts interconnect.