Finding Trigonometric Functions Given Secant And Sine Sign

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In trigonometry, determining the values of all six trigonometric functions is a common task. Often, we are given information about one or two trigonometric functions and the quadrant in which the angle lies. This allows us to use trigonometric identities and relationships to find the remaining functions. In this article, we'll explore a step-by-step approach to finding the five remaining trigonometric functions when given the secant of an angle and the sign of its sine.

Problem Statement

Let's consider the following problem:

Find the five remaining trigonometric functions of θ, given that:

  • sec θ = 4/3
  • sin θ < 0

Complete the following table:

sin θ = ?
cos θ = 3/4
tan θ = ?
csc θ = ?
sec θ = 4/3
cot θ = ?

Step-by-Step Solution

1. Identify Given Information

The first step is to clearly identify the information provided. We are given that:

  • The secant of θ (sec θ) is 4/3. Secant is the reciprocal of the cosine function, meaning sec θ = 1/cos θ.
  • The sine of θ (sin θ) is negative. This tells us that the angle θ lies in either the third or fourth quadrant, where the sine function is negative.

2. Determine Cosine

Leveraging the Reciprocal Identity: Given that sec θ is the reciprocal of cos θ, we can easily find the cosine of θ:

cos θ = 1 / sec θ = 1 / (4/3) = 3/4

So, we have cos θ = 3/4. This confirms the value provided in the table.

3. Determine the Quadrant

Combining Information: We know that:

  • cos θ = 3/4 (positive)
  • sin θ < 0 (negative)

The cosine function is positive in the first and fourth quadrants, while the sine function is negative in the third and fourth quadrants. The quadrant where both conditions are met is the fourth quadrant. Therefore, θ lies in the fourth quadrant.

4. Determine Sine

Using the Pythagorean Identity: The fundamental Pythagorean identity in trigonometry is:

sin² θ + cos² θ = 1

We can use this identity to find sin θ. We already know cos θ = 3/4, so:

sin² θ + (3/4)² = 1 sin² θ + 9/16 = 1 sin² θ = 1 - 9/16 sin² θ = 7/16 sin θ = ±√(7/16) sin θ = ±√7 / 4

Since we know that sin θ < 0 (θ is in the fourth quadrant), we take the negative root:

sin θ = -√7 / 4

5. Determine Tangent

Using the Quotient Identity: The tangent function is defined as the ratio of sine to cosine:

tan θ = sin θ / cos θ

Substituting the values we found:

tan θ = (-√7 / 4) / (3/4) tan θ = (-√7 / 4) * (4/3) tan θ = -√7 / 3

6. Determine Cosecant

Leveraging the Reciprocal Identity: Cosecant (csc θ) is the reciprocal of sine (sin θ):

csc θ = 1 / sin θ

Substituting the value of sin θ:

csc θ = 1 / (-√7 / 4) csc θ = -4 / √7

To rationalize the denominator, we multiply the numerator and denominator by √7:

csc θ = (-4 / √7) * (√7 / √7) csc θ = -4√7 / 7

7. Determine Cotangent

Leveraging the Reciprocal Identity: Cotangent (cot θ) is the reciprocal of tangent (tan θ):

cot θ = 1 / tan θ

Substituting the value of tan θ:

cot θ = 1 / (-√7 / 3) cot θ = -3 / √7

To rationalize the denominator, we multiply the numerator and denominator by √7:

cot θ = (-3 / √7) * (√7 / √7) cot θ = -3√7 / 7

Summary of Results

We have now found all five remaining trigonometric functions:

  • sin θ = -√7 / 4
  • cos θ = 3/4 (Given)
  • tan θ = -√7 / 3
  • csc θ = -4√7 / 7
  • sec θ = 4/3 (Given)
  • cot θ = -3√7 / 7

Completing the table, we have:

sin θ = -√7 / 4
cos θ = 3/4
tan θ = -√7 / 3
csc θ = -4√7 / 7
sec θ = 4/3
cot θ = -3√7 / 7

Key Trigonometric Identities Used

Throughout the solution, we utilized several key trigonometric identities:

  1. Reciprocal Identities:
    • sec θ = 1 / cos θ
    • csc θ = 1 / sin θ
    • cot θ = 1 / tan θ
  2. Quotient Identity:
    • tan θ = sin θ / cos θ
  3. Pythagorean Identity:
    • sin² θ + cos² θ = 1

Importance of Quadrant Information

The information about the sign of sin θ (sin θ < 0) was crucial in determining the correct quadrant for the angle θ. Without this information, we would have had two possible solutions for sin θ (±√7 / 4), and we wouldn't have been able to choose the correct one. The quadrant information helps us narrow down the possibilities and find the unique values of the trigonometric functions.

Conclusion

Finding the remaining trigonometric functions when given some initial information is a fundamental problem in trigonometry. By using trigonometric identities, reciprocal relationships, and quadrant information, we can systematically determine the values of all six trigonometric functions. In this article, we demonstrated how to find the five remaining trigonometric functions when given the secant of an angle and the sign of its sine. Understanding these concepts is essential for further studies in mathematics, physics, and engineering.

By mastering these techniques, students and enthusiasts can confidently tackle a wide range of trigonometric problems and gain a deeper understanding of the relationships between trigonometric functions. Remember to always consider the given information carefully and apply the appropriate identities to arrive at the correct solution. The trigonometric functions are vital in understanding various mathematical and real-world phenomena.