Problem 20 - Precalculus with Limits

Question

In Exercises 11 - 24, use the given values to evaluate (if possible)all six trigonometric functions. \( \sec x = 4 \), \( \sin x > 0 \)

Step-by-Step Solution

Answer

\(\sin x = 0.9682458366\), \(\cos x = 0.25\), \(\tan x = 3.872983346\), \(\csc x = 1.032795559\), \(\sec x = 4\), \(\cot x = 0.2581988897\)

Step 1: Calculate the value of \(\cos x\) using \(\sec x\)

The secant of an angle in a right triangle is the reciprocal of the cosine. Therefore, \(\cos x = \frac{1}{\sec x} = \frac{1}{4} = 0.25\).

Step 2: Calculate the value of \(\sin x\) using the Pythagorean identity

The Pythagorean identity is \(\sin^2 x + \cos^2 x = 1\). Plugging in the value from Step 1, we find \(\sin^2 x = 1 - \cos^2 x = 1 - (0.25)^2 = 0.9375\). Since we know from the problem that \(\sin x > 0\), we have \(\sin x = \sqrt{0.9375} = 0.9682458366\), approximately.

Step 3: Calculate the value of \(\tan x\) using \(\sin x\) and \(\cos x\)

The tangent of an angle in a right triangle is the ratio of the sine to the cosine. Therefore, \(\tan x = \frac{\sin x}{\cos x} = \frac{0.9682458366}{0.25} = 3.872983346\), approximately.

Step 4: Calculate the value of \(\csc x\), \(\sec x\) and \(\cot x\)

These are reciprocals of \(\sin x\), \(\cos x\) and \(\tan x\) respectively. So, \(\csc x = \frac{1}{\sin x} = 1.032795559\), \(\sec x = 4\) (given in the problem), \(\cot x = \frac{1}{\tan x} = 0.2581988897\), approximately.

Key Concepts

Secant Function

The secant function is one of the six fundamental trigonometric functions. It is denoted as \( \text{sec}(x) \) and is defined as the reciprocal of the cosine function, meaning \( \text{sec}(x) = \frac{1}{\text{cos}(x)} \).

When dealing with a right-angled triangle, the secant of an angle represents the ratio of the length of the hypotenuse to the length of the adjacent side. It's important for students to remember that a secant value, such as \( \text{sec}(x) = 4 \) provided in the exercise, tells us immediately that \( \text{cos}(x) = \frac{1}{4} \) without the need for further calculations.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry that links the square of the sine and cosine of an angle. The identity states that for any angle \( x \), \( \text{sin}^2(x) + \text{cos}^2(x) = 1 \).

This relationship is derived from the Pythagorean theorem and it is valid for all real numbers \( x \). In our exercise, once the value of cosine has been established, we can use the Pythagorean identity to find the sine value. Because \( \text{sin}(x) > 0 \), we use the positive square root of \( 0.9375 \) to obtain an approximate sine value. Remember to always consider the context of the problem to decide whether the positive or negative square root should be used.

Tangent Function

The tangent function, represented as \( \text{tan}(x) \), is another key trigonometric function. It is defined as the ratio of the sine of an angle to its cosine, \( \text{tan}(x) = \frac{\text{sin}(x)}{\text{cos}(x)} \).

In the context of right-angled triangles, it corresponds to the ratio of the side opposite to the angle over the side adjacent to it. As we've established the values for both sine and cosine from our exercise, finding the tangent is straightforward. The tangent value can help us understand the angle's slope, as it physically represents the slope of the line created when we plot the angle on a coordinate grid. Because tangent is sine divided by cosine, any errors in the earlier steps could result in a larger error in the tangent value, showing the importance of accuracy in calculations.