Finding Exact Values Of Cos Θ, Sec Θ, And Cot Θ On The Unit Circle

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In trigonometry, understanding the relationship between angles and points on the unit circle is fundamental. This article delves into how to find the exact values of trigonometric functions, specifically cosine, secant, and cotangent, when given a point on the unit circle that corresponds to the terminal side of an angle in standard position.

Understanding the Unit Circle and Trigonometric Functions

The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system. Its equation is given by x2 + y2 = 1. Any point on the unit circle can be represented by coordinates (x, y), where x and y are related to trigonometric functions of an angle θ, which is measured counterclockwise from the positive x-axis.

Cosine and Secant

The cosine of an angle θ, denoted as cos θ, is defined as the x-coordinate of the point where the terminal side of the angle intersects the unit circle. Mathematically, cos θ = x. The secant of an angle θ, denoted as sec θ, is the reciprocal of the cosine function. Thus, sec θ = 1 / cos θ = 1 / x, provided that x ≠ 0. Secant represents the ratio of the hypotenuse to the adjacent side in a right triangle representation of the angle.

Cotangent

The cotangent of an angle θ, denoted as cot θ, is the reciprocal of the tangent function. The tangent function is defined as tan θ = y / x, so cot θ = 1 / tan θ = x / y, provided that y ≠ 0. Cotangent can be seen as the ratio of the adjacent side to the opposite side in the right triangle context.

Problem Statement

Suppose that θ is an angle in standard position whose terminal side intersects the unit circle at the point (-12/37, -35/37). Our goal is to find the exact values of cos θ, sec θ, and cot θ.

Step-by-Step Solution

  1. Identify the Coordinates: The point of intersection is given as (-12/37, -35/37). Here, x = -12/37 and y = -35/37.

  2. Find cos θ:

    • Since cos θ = x, we have cos θ = -12/37. This is the exact value of the cosine of the angle θ.

  3. Find sec θ:

    • The secant is the reciprocal of the cosine, so sec θ = 1 / cos θ.
    • Substituting the value of cos θ, we get sec θ = 1 / (-12/37) = -37/12. This is the exact value of the secant of the angle θ.

  4. Find cot θ:

    • The cotangent is the reciprocal of the tangent, where cot θ = x / y.
    • Substituting the values of x and y, we have cot θ = (-12/37) / (-35/37).
    • Simplifying the fraction, cot θ = (-12/37) * (-37/35) = 12/35. This gives us the exact value of the cotangent of the angle θ.

Detailed Explanation

Understanding Cosine (cos θ)

In the realm of trigonometry, cosine serves as a fundamental function, intimately linked to the unit circle. When an angle θ is positioned in standard form, with its vertex at the origin and initial side along the positive x-axis, its terminal side intersects the unit circle at a unique point. The x-coordinate of this intersection point precisely defines the cosine of the angle. In mathematical notation, this is expressed as cos θ = x. For the given problem, the terminal side of the angle θ intersects the unit circle at the point (-12/37, -35/37). Therefore, the x-coordinate, which is -12/37, gives us the cosine of the angle. Thus, cos θ = -12/37.

This concept is crucial because it establishes a direct geometric interpretation of cosine. By understanding that cosine is the x-coordinate on the unit circle, we can quickly determine its value for any angle whose terminal side's intersection point is known. Moreover, this understanding is vital in numerous applications, ranging from physics to engineering, where angles and their trigonometric functions are used to model periodic phenomena, oscillations, and wave behavior. The negative value of cosine in this instance indicates that the angle θ lies in either the second or third quadrant, where x-coordinates are negative.

Deep Dive into Secant (sec θ)

The secant function, denoted as sec θ, is defined as the reciprocal of the cosine function. In other words, sec θ = 1 / cos θ. This reciprocal relationship makes secant particularly useful in situations where cosine is in the denominator, simplifying calculations and offering a different perspective on the angle's properties. For our specific problem, having already determined that cos θ = -12/37, we can find sec θ by taking the reciprocal of -12/37. This yields sec θ = 1 / (-12/37) = -37/12. The negative sign arises directly from the negative value of the cosine, as secant will have the same sign as cosine.

Delving deeper into the significance of secant, it's important to recognize its role in right-angled triangles. If we consider a right triangle formed by the x-axis, the terminal side of θ, and a perpendicular line to the x-axis, sec θ represents the ratio of the hypotenuse to the adjacent side. This interpretation is crucial in various applications, such as navigation and surveying, where angles and distances are often analyzed using trigonometric ratios. Moreover, in calculus and advanced mathematics, secant appears frequently in integrals and differential equations, highlighting its importance beyond basic trigonometry. Understanding secant not just as a reciprocal of cosine but as a distinct trigonometric function with its own geometric and analytical properties is key to mastering trigonometric concepts.

Exploring Cotangent (cot θ) in Detail

The cotangent function, abbreviated as cot θ, is defined as the reciprocal of the tangent function. Since tangent (tan θ) is defined as the ratio of the sine to the cosine (tan θ = y / x), cotangent is the ratio of the cosine to the sine (cot θ = x / y). In the context of the unit circle, cot θ is derived from the coordinates of the point where the terminal side of the angle intersects the circle. For the given problem, the intersection point is (-12/37, -35/37), which means x = -12/37 and y = -35/37. Thus, cot θ = x / y = (-12/37) / (-35/37).

To find the exact value, we divide -12/37 by -35/37. Dividing by a fraction is equivalent to multiplying by its reciprocal, so we have cot θ = (-12/37) * (-37/35). The negative signs cancel each other out, and the 37s also cancel, simplifying the expression to cot θ = 12/35. This positive value indicates that the angle θ lies in either the first or third quadrant, where both x and y have the same sign (both negative in this case). Cotangent’s role in trigonometry extends to its representation in right triangles, where it is the ratio of the adjacent side to the opposite side. This geometric interpretation is highly valuable in applications such as land surveying, where angles of elevation and depression are crucial, and in complex number theory, where cotangent functions appear in various identities and transformations. Additionally, in calculus, cotangent functions arise in the integration of certain trigonometric expressions, further underscoring its importance.

Conclusion

In summary, given the point (-12/37, -35/37) on the unit circle corresponding to the terminal side of angle θ, we found that cos θ = -12/37, sec θ = -37/12, and cot θ = 12/35. These exact values are essential in various mathematical and scientific applications. Understanding the definitions and relationships between trigonometric functions and the unit circle allows for precise calculations and a deeper comprehension of angular relationships. This problem illustrates the fundamental principles of trigonometry and their practical application in determining exact trigonometric values.

By mastering these concepts, students and professionals alike can confidently tackle more complex problems involving trigonometric functions and their applications in various fields.