Calculate Cos 870 Degrees A Step-by-Step Trigonometry Guide

The question asks us to calculate the cosine of 870 degrees. This falls under the category of trigonometry within algebra, where we deal with angles and their trigonometric functions. To effectively solve this, we'll break down the problem into manageable steps, utilizing the periodic nature of trigonometric functions and reference angles.

Understanding Trigonometric Functions and Periodicity

Trigonometric functions, including cosine, are periodic. This means their values repeat after a certain interval. For the cosine function, the period is 360 degrees. This property is crucial because it allows us to simplify angles by adding or subtracting multiples of 360 degrees without changing the value of the trigonometric function. In simpler terms, cos(θ) = cos(θ + 360k), where k is any integer. This is because after every full rotation (360 degrees), the angle returns to its original position on the unit circle, hence the cosine value remains the same.

Understanding the unit circle is fundamental to grasping trigonometric functions. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Angles are measured counterclockwise from the positive x-axis. The cosine of an angle is represented by the x-coordinate of the point where the terminal side of the angle intersects the unit circle, while the sine is represented by the y-coordinate. The periodicity of trigonometric functions stems from the fact that as you rotate around the unit circle, you eventually return to the same points, thus repeating the x and y coordinates, and consequently, the cosine and sine values.

The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It helps in finding the trigonometric values of angles in different quadrants. To find the reference angle, you first need to determine the quadrant in which the angle lies. Then, you subtract the nearest multiple of 180 degrees from the angle (if it's in the second or third quadrant) or subtract the angle from 360 degrees (if it's in the fourth quadrant). The reference angle simplifies the calculation because the trigonometric values of angles in different quadrants are related to the trigonometric values of their reference angles. The sign (+ or -) is determined by the quadrant in which the original angle lies.

Step-by-Step Solution for cos 870°

To find cos 870°, we first need to reduce the angle to an equivalent angle within the range of 0 to 360 degrees. We can achieve this by subtracting multiples of 360 degrees from 870 degrees:

870° - 360° = 510°

510° - 360° = 150°

So, cos 870° = cos 150°. Now, we need to find the reference angle for 150 degrees. Since 150 degrees lies in the second quadrant, the reference angle is:

180° - 150° = 30°

Thus, the reference angle is 30 degrees. We know that cos 30° = √3/2. However, since 150 degrees is in the second quadrant, where cosine values are negative, we have:

cos 150° = -cos 30° = -√3/2

Therefore, cos 870° = -√3/2.

Detailed Explanation of the Solution Process

To recap, the initial step in solving cos 870° involves reducing the angle to an equivalent angle within the range of 0° to 360°. We achieve this by subtracting multiples of 360° until we get an angle within this range. The periodicity of the cosine function allows us to do this without altering the cosine value. Trigonometric functions repeat their values after every full rotation around the unit circle, which is 360 degrees.

Once we have the reduced angle (150° in this case), the next step is to find the reference angle. The reference angle is the acute angle formed between the terminal side of the angle and the x-axis. It helps simplify the calculation by relating the cosine value of the angle to the cosine value of an acute angle. For an angle in the second quadrant (like 150°), the reference angle is calculated by subtracting the angle from 180°.

After finding the reference angle (30°), we determine the cosine of the reference angle, which is cos 30° = √3/2. However, we must consider the quadrant in which the original angle lies. In the second quadrant, cosine values are negative. This is because, in the second quadrant, the x-coordinates (which represent cosine values on the unit circle) are negative.

Finally, we apply the appropriate sign to the cosine value of the reference angle. Since 150° is in the second quadrant, the cosine value is negative. Thus, cos 150° = -cos 30° = -√3/2. This gives us the final answer: cos 870° = -√3/2.

Common Mistakes and How to Avoid Them

Several common mistakes can occur when calculating trigonometric functions of large angles. One frequent error is forgetting to reduce the angle to its equivalent within the 0° to 360° range. Without this step, it becomes difficult to determine the reference angle and the correct sign of the trigonometric function.

Another common mistake is incorrectly calculating the reference angle. It's essential to remember the rules for each quadrant. For angles in the second quadrant, subtract the angle from 180°; for angles in the third quadrant, subtract 180° from the angle; and for angles in the fourth quadrant, subtract the angle from 360°. A clear understanding of these rules is critical for accurate calculations.

Perhaps the most pervasive mistake is overlooking the sign of the trigonometric function in the appropriate quadrant. Remember the mnemonic "All Students Take Calculus" (ASTC) to help recall which trigonometric functions are positive in each quadrant: All (all trigonometric functions are positive in the first quadrant), Students (sine is positive in the second quadrant), Take (tangent is positive in the third quadrant), Calculus (cosine is positive in the fourth quadrant). Make sure to apply the correct sign based on the quadrant in which the angle lies.

To avoid these mistakes, it is helpful to practice a variety of similar problems. Regularly working through such problems will solidify your understanding of the steps involved and reduce the likelihood of errors. Additionally, drawing a unit circle and marking the angle and its reference angle can provide a visual aid that helps to ensure accuracy. Double-checking your calculations and paying careful attention to the quadrant and the sign are also crucial steps in preventing errors.

Conclusion

The correct answer is C) -√3/2. By understanding the periodicity of trigonometric functions, using reference angles, and considering the signs in different quadrants, we can accurately calculate trigonometric values for any angle. Mastering these concepts is essential for success in trigonometry and related fields. Remember to practice regularly and pay attention to detail to avoid common mistakes.

By following these detailed steps and explanations, you can confidently calculate the cosine of 870 degrees and similar trigonometric problems. Trigonometry is a fundamental aspect of mathematics, and a solid understanding of its principles is invaluable for various applications in science and engineering.

Final Answer: The final answer is C) -√3/2