Trigonometric Functions Calculation Given Cosine Value

When given the cosine value of an angle θ, we can determine the exact values of the other trigonometric functions. In this article, we will explore a step-by-step approach to finding these values, including simplifying radicals and using integers or fractions in our expressions. This comprehensive guide will walk you through the process, ensuring a clear understanding of the methodology involved.

Understanding the Given Information

In this specific case, we are given that cos θ = -4√3 / 7. This information is crucial because it tells us both the ratio of the adjacent side to the hypotenuse in a right triangle and the quadrant in which the angle θ lies. Since the cosine value is negative, θ must be in either the second or third quadrant, where the x-coordinate (adjacent side) is negative. To determine the other trigonometric function values, we need to find the sine and tangent of θ, as well as their reciprocals: cosecant, secant, and cotangent.

Understanding the given cosine value is the first step in determining the other trigonometric function values. The negative sign indicates that the angle θ is located in either the second or third quadrant. Cosine represents the ratio of the adjacent side to the hypotenuse in a right triangle. To find the other trigonometric functions, we need to determine the sine and tangent of θ, along with their reciprocals: cosecant, secant, and cotangent. To proceed, we can use the Pythagorean identity sin² θ + cos² θ = 1 to find the value of sin θ. This identity is a fundamental relationship between sine and cosine and will help us uncover the missing piece of information needed to complete our trigonometric picture. By substituting the given cosine value into this identity, we can solve for sin θ and subsequently find the other trigonometric functions.

Finding the Sine Value

To find the sine value, we utilize the Pythagorean identity:

sin² θ + cos² θ = 1

Substitute the given cosine value:

sin² θ + (-4√3 / 7)² = 1

Calculate the square:

sin² θ + (16 * 3) / 49 = 1

sin² θ + 48 / 49 = 1

Isolate sin² θ:

sin² θ = 1 - 48 / 49

sin² θ = 1 / 49

Take the square root:

sin θ = ±√(1 / 49)

sin θ = ±1 / 7

Since θ is in either the second or third quadrant, we need to determine the sign of sin θ. In the second quadrant, sine is positive, while in the third quadrant, sine is negative. Without additional information to specify the quadrant, we will consider both possibilities. However, for the sake of demonstration, let's assume θ is in the second quadrant, where sin θ = 1 / 7. This assumption allows us to proceed with a specific value for sine and find the remaining trigonometric functions.

Finding the sine value requires the use of the Pythagorean identity, a cornerstone of trigonometric relationships. Substituting the given cosine value into the identity allows us to isolate sin² θ and eventually solve for sin θ. The calculation involves squaring the cosine value, simplifying fractions, and taking the square root. This process yields two possible values for sin θ: a positive and a negative value. The sign of sin θ depends on the quadrant in which θ lies. Since cosine is negative, θ is in either the second or third quadrant. In the second quadrant, sine is positive, while in the third quadrant, sine is negative. Without further information to pinpoint the exact quadrant, both possibilities must be considered. However, for the purpose of this illustration, we will assume θ is in the second quadrant, leading to sin θ = 1 / 7. This assumption allows us to continue with a specific value and determine the other trigonometric functions.

Calculating Tangent

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

tan θ = sin θ / cos θ

Substitute the values of sine and cosine:

tan θ = (1 / 7) / (-4√3 / 7)

Simplify the expression:

tan θ = (1 / 7) * (7 / -4√3)

tan θ = 1 / -4√3

Rationalize the denominator by multiplying the numerator and denominator by √3:

tan θ = (1 * √3) / (-4√3 * √3)

tan θ = √3 / (-4 * 3)

tan θ = -√3 / 12

Thus, the exact value of tan θ is -√3 / 12. This value is obtained by dividing the sine by the cosine and simplifying the resulting fraction. Rationalizing the denominator ensures the expression is in its simplest form, making it easier to work with in subsequent calculations or applications.

The tangent function, defined as the ratio of sine to cosine, is a key trigonometric function. To calculate tan θ, we substitute the values of sine and cosine that we have already determined. This involves dividing a fraction by another fraction, which can be simplified by multiplying by the reciprocal. The resulting expression often involves a radical in the denominator, which is not considered simplified. To rationalize the denominator, we multiply both the numerator and the denominator by the radical. This process eliminates the radical from the denominator, resulting in a simplified expression for tan θ. The final value, -√3 / 12, represents the exact tangent of the angle θ, given the initial cosine value and the assumption that θ is in the second quadrant. This calculation demonstrates the importance of simplifying radicals and rationalizing denominators in trigonometric problems.

Finding the Reciprocal Functions

Now that we have the values for sine, cosine, and tangent, we can find their reciprocal functions:

• Cosecant (csc θ): The reciprocal of sine. csc θ = 1 / sin θ = 1 / (1 / 7) = 7

• Secant (sec θ): The reciprocal of cosine. sec θ = 1 / cos θ = 1 / (-4√3 / 7) = 7 / -4√3

  Rationalize the denominator:

  sec θ = (7 * √3) / (-4√3 * √3) = 7√3 / (-4 * 3) = -7√3 / 12

• Cotangent (cot θ): The reciprocal of tangent. cot θ = 1 / tan θ = 1 / (-√3 / 12) = -12 / √3

  Rationalize the denominator:

  cot θ = (-12 * √3) / (√3 * √3) = -12√3 / 3 = -4√3

Therefore, the reciprocal functions are: csc θ = 7, sec θ = -7√3 / 12, and cot θ = -4√3. These values are obtained by taking the reciprocals of the sine, cosine, and tangent values, respectively. Rationalizing the denominator is again necessary to present the secant and cotangent in their simplest forms. This step completes the process of finding all six trigonometric functions for the given angle θ.

Finding the reciprocal functions involves taking the inverse of the sine, cosine, and tangent values. This process is straightforward but often requires additional simplification, particularly when radicals are involved. Cosecant, the reciprocal of sine, is simply the inverse of sin θ. Secant, the reciprocal of cosine, requires rationalizing the denominator after taking the inverse. Similarly, cotangent, the reciprocal of tangent, also needs to be rationalized. Rationalizing the denominator ensures that the expressions are in their simplest and most standard form. The resulting values for the reciprocal functions, csc θ = 7, sec θ = -7√3 / 12, and cot θ = -4√3, provide a complete set of trigonometric values for the angle θ, given the initial cosine value and the assumption of the second quadrant. This comprehensive approach highlights the importance of both reciprocal relationships and simplification techniques in trigonometry.

Summary of Trigonometric Function Values

Given cos θ = -4√3 / 7 (and assuming θ is in the second quadrant), we have found the following trigonometric function values:

• sin θ = 1 / 7
• cos θ = -4√3 / 7
• tan θ = -√3 / 12
• csc θ = 7
• sec θ = -7√3 / 12
• cot θ = -4√3

This comprehensive set of values provides a complete picture of the trigonometric functions for the angle θ. The process involved using the Pythagorean identity to find sine, calculating tangent as the ratio of sine to cosine, and determining the reciprocal functions by taking the inverses. Rationalizing the denominator was a crucial step in simplifying the expressions for tangent, secant, and cotangent. This example illustrates a systematic approach to finding all trigonometric function values when one value is given, emphasizing the importance of trigonometric identities, reciprocal relationships, and simplification techniques.

In summary, determining the trigonometric function values when given the cosine involves a series of steps that utilize fundamental trigonometric identities and algebraic manipulations. The Pythagorean identity allows us to find the sine value, and the tangent is calculated as the ratio of sine to cosine. The reciprocal functions are then found by taking the inverses of sine, cosine, and tangent. Rationalizing the denominator is a critical step in simplifying the expressions and presenting them in their standard form. This process not only provides the exact values of all six trigonometric functions but also reinforces the interconnectedness of these functions and the importance of algebraic simplification in trigonometric problems. The systematic approach outlined in this article can be applied to similar problems, ensuring a thorough understanding of trigonometric relationships and problem-solving techniques.