Problem 26 - Precalculus with Calculus Previews

Question

Use the given information to find the values of the remaining five trigonometric functions. $$ \sin x=\frac{4}{5}, \quad 0

Step-by-Step Solution

Answer

\( \sin x = \frac{4}{5} \), \( \cos x = \frac{3}{5} \), \( \tan x = \frac{4}{3} \), \( \cot x = \frac{3}{4} \), \( \sec x = \frac{5}{3} \), \( \csc x = \frac{5}{4} \).

Step 1: Recall the Identity

We know that the main Pythagorean identity for sine and cosine is \( \sin^2 x + \cos^2 x = 1 \). We will use this to find the cosine of angle \( x \).

Step 2: Calculate \( \cos x \)

Given \( \sin x = \frac{4}{5} \), substitute this into the identity: \[ \left(\frac{4}{5}\right)^2 + \cos^2 x = 1 \]Simplifying, we get: \[ \frac{16}{25} + \cos^2 x = 1 \]Subtract \( \frac{16}{25} \) from 1: \[ \cos^2 x = 1 - \frac{16}{25} = \frac{9}{25} \] Therefore, \( \cos x = \frac{3}{5} \) since \( 0 < x < \pi/2 \) (cosine is positive in the first quadrant).

Step 3: Find \( \tan x \)

\( \tan x = \frac{\sin x}{\cos x} \).Using the calculated values: \[ \tan x = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3} \]

Step 4: Find \( \cot x \)

\( \cot x \) is the reciprocal of \( \tan x \):\[ \cot x = \frac{1}{\tan x} = \frac{3}{4} \]

Step 5: Find \( \sec x \)

\( \sec x \) is the reciprocal of \( \cos x \): \[ \sec x = \frac{1}{\cos x} = \frac{5}{3} \]

Step 6: Find \( \csc x \)

\( \csc x \) is the reciprocal of \( \sin x \): \[ \csc x = \frac{1}{\sin x} = \frac{5}{4} \]

Key Concepts

Pythagorean Identity

The Pythagorean identity is one of the fundamental relationships in trigonometry that ties together the sine and cosine of an angle. This identity is represented by the equation \( \sin^2 x + \cos^2 x = 1 \). It is called the "Pythagorean" identity because it resembles the Pythagorean theorem in geometry, which deals with the squares of the lengths of a right triangle's sides.
When you are given the sine of an angle, you can use this identity to find the cosine of the same angle and vice versa. For example, if \( \sin x = \frac{4}{5} \), you can substitute into the identity:

• Calculate \( \sin^2 x = \left(\frac{4}{5}\right)^2 = \frac{16}{25}\).
• Substitute \( \sin^2 x \) into the equation: \( \frac{16}{25} + \cos^2 x = 1 \).
• Solve for \( \cos^2 x \): \( \cos^2 x = 1 - \frac{16}{25} = \frac{9}{25} \).
• Therefore, \( \cos x = \frac{3}{5} \) since cosine is positive in the first quadrant.

This identity is particularly useful because it allows you to move seamlessly between sine and cosine, revealing how interconnected these trigonometric functions are.

Sine and Cosine Relationship

The sine and cosine functions are closely related to each other and describe the ratios of the sides of a right triangle relative to an angle. Specifically, these functions provide a way to understand angles in a circle and convert between different trigonometric functions.
The sine function, denoted as \( \sin x \), represents the opposite side over the hypotenuse in a right triangle. Meanwhile, the cosine function, \( \cos x \), gives you the adjacent side over the hypotenuse. These definitions are pivotal because they imply that knowing one function often enables determination of the other through geometric or algebraic means.
In our example, given \( \sin x = \frac{4}{5} \), the Pythagorean identity helps to calculate \( \cos x \). This connection demonstrates how these two functions serve complementary roles in trigonometry, often leading to other trigonometric insights.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions: sine, cosine, and tangent. These reciprocals include cosecant, secant, and cotangent.
Here's a quick breakdown:

• Cosecant (\( \csc x \)) is the reciprocal of sine, given by \( \csc x = \frac{1}{\sin x} \).
• Secant (\( \sec x \)) is the reciprocal of cosine, expressed as \( \sec x = \frac{1}{\cos x} \).
• Cotangent (\( \cot x \)) is the reciprocal of tangent, defined as \( \cot x = \frac{1}{\tan x} \).

By understanding these reciprocals, you can quickly find the value of one function if you know its corresponding basic function.
For example, knowing \( \sin x = \frac{4}{5} \) allows you to calculate \( \csc x = \frac{5}{4} \). Similarly, knowing \( \cos x = \frac{3}{5} \) leads to \( \sec x = \frac{5}{3} \). And using \( \tan x = \frac{4}{3} \), you find \( \cot x = \frac{3}{4} \).
Understanding these connections makes solving trigonometric problems more manageable and helps deepen comprehension of the relationships between these functions.