Finding The Value Of Cos(-690 Degrees) A Step-by-Step Guide

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Understanding trigonometric functions and their values at various angles is a fundamental aspect of algebra and trigonometry. This article delves into the process of finding the value of cos(-690°), providing a step-by-step explanation to ensure clarity and comprehension. We will explore the concepts of coterminal angles, reference angles, and the unit circle to arrive at the correct answer. By the end of this guide, you will not only know the value of cos(-690°) but also grasp the underlying principles that govern trigonometric function evaluations.

Understanding Trigonometric Functions and Angles

Before diving into the specifics of cos(-690°), it's essential to have a firm grasp of trigonometric functions and how angles are measured in trigonometry. Trigonometric functions, such as sine (sin), cosine (cos), and tangent (tan), relate the angles of a right triangle to the ratios of its sides. These functions are periodic, meaning their values repeat after a certain interval. For cosine, the period is 360°, which implies that cos(θ) = cos(θ + 360°k) for any integer k. This property is crucial for simplifying angles that are outside the standard range of 0° to 360°.

Angles in trigonometry are typically measured in degrees or radians. A full circle is 360 degrees or 2π radians. Negative angles represent a clockwise rotation from the positive x-axis, while positive angles represent a counterclockwise rotation. Understanding the unit circle, a circle with a radius of 1 centered at the origin, is also critical. The x-coordinate of a point on the unit circle corresponds to the cosine of the angle, and the y-coordinate corresponds to the sine of the angle. Mastering these concepts lays the groundwork for accurately determining trigonometric values for any angle.

Finding Coterminal Angles

The concept of coterminal angles is vital when dealing with angles outside the range of 0° to 360° or angles that are negative. Coterminal angles are angles that share the same initial and terminal sides. To find a coterminal angle, you can add or subtract multiples of 360° (or 2π radians) from the given angle. This process simplifies the problem by bringing the angle within a more manageable range.

In the case of cos(-690°), the angle -690° is a negative angle, indicating a clockwise rotation. To find a coterminal angle within the range of 0° to 360°, we can add multiples of 360° to -690° until we obtain an angle within this range. Adding 360° once gives us -690° + 360° = -330°, which is still negative. Adding 360° again gives us -330° + 360° = 30°. Therefore, -690° and 30° are coterminal angles. This means that cos(-690°) is equal to cos(30°). Identifying coterminal angles is a crucial step in simplifying trigonometric problems, as it allows us to work with angles that are easier to visualize and evaluate. The ability to quickly find coterminal angles is a valuable skill in trigonometry, enabling us to solve complex problems with greater efficiency and accuracy.

Determining the Reference Angle

Once we have a coterminal angle within the range of 0° to 360°, the next step is to determine the reference angle. The reference angle is the acute angle (an angle between 0° and 90°) formed by the terminal side of the angle and the x-axis. Reference angles help simplify trigonometric calculations because the trigonometric functions of an angle and its reference angle have the same absolute values. The only difference might be the sign, which depends on the quadrant in which the angle lies.

For the coterminal angle of 30°, the reference angle is simply 30° because it already lies in the first quadrant (0° to 90°). In the first quadrant, all trigonometric functions (sine, cosine, tangent) are positive. If the angle were in a different quadrant, we would calculate the reference angle differently. For example, in the second quadrant (90° to 180°), the reference angle is calculated as 180° - θ; in the third quadrant (180° to 270°), it's θ - 180°; and in the fourth quadrant (270° to 360°), it's 360° - θ. However, since our coterminal angle is 30°, the reference angle is straightforward.

Understanding reference angles is a fundamental skill in trigonometry, enabling us to break down complex angles into simpler, more manageable forms. By identifying the reference angle and the quadrant in which the angle lies, we can easily determine the sign and value of any trigonometric function.

Evaluating cos(30°) Using the Unit Circle

Now that we have the reference angle of 30°, we can evaluate cos(30°) using our knowledge of the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Points on the unit circle are defined by their coordinates (cos θ, sin θ), where θ is the angle formed by the positive x-axis and the line connecting the origin to the point.

For an angle of 30°, the corresponding point on the unit circle is (√3/2, 1/2). The x-coordinate of this point represents the cosine of 30°, and the y-coordinate represents the sine of 30°. Therefore, cos(30°) = √3/2. This value is a standard trigonometric value that is important to memorize or be able to quickly derive. Knowing the unit circle and the coordinates for common angles such as 0°, 30°, 45°, 60°, and 90° greatly simplifies trigonometric calculations.

Alternatively, we can also derive this value using a 30-60-90 triangle. In a 30-60-90 triangle, the sides are in the ratio 1:√3:2, where the side opposite the 30° angle is 1, the side opposite the 60° angle is √3, and the hypotenuse is 2. The cosine of 30° is the adjacent side divided by the hypotenuse, which is √3/2. This method provides a geometric understanding of the value and reinforces the relationship between trigonometric functions and right triangles.

Final Answer: The Value of cos(-690°)

Having followed the steps of finding the coterminal angle, determining the reference angle, and evaluating the cosine of the reference angle, we can now confidently state the value of cos(-690°). As we established, cos(-690°) is equal to cos(30°), and cos(30°) is √3/2.

Therefore, the value of cos(-690°) is √3/2. This result highlights the importance of understanding trigonometric properties and applying them systematically to solve problems. By breaking down the problem into smaller, more manageable steps, we can accurately determine trigonometric values for any angle.

In summary, the process involves first finding a coterminal angle within the range of 0° to 360°, then determining the reference angle, and finally using the unit circle or trigonometric ratios of special triangles to find the value of the trigonometric function. This method can be applied to evaluate any trigonometric function for any angle, making it a crucial skill in mathematics and related fields.

Conclusion

In conclusion, finding the value of cos(-690°) involves understanding coterminal angles, reference angles, and the unit circle. By adding multiples of 360° to -690°, we find the coterminal angle of 30°. The reference angle is also 30°, and using the unit circle or 30-60-90 triangle properties, we determine that cos(30°) = √3/2. Therefore, cos(-690°) = √3/2. This detailed explanation illustrates the systematic approach to solving trigonometric problems and reinforces the fundamental concepts of trigonometry. Mastering these concepts is essential for success in advanced mathematics and related disciplines.