Calculating Sin(-135°) Using Reference Angles A Step-by-Step Guide

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Determining the exact trigonometric values for angles, especially those outside the standard 0° to 90° range, often requires a systematic approach. In this article, we will delve into a detailed, step-by-step method to calculate the exact value of sin(-135°). This exploration will not only help you understand the specific solution but also equip you with a general strategy applicable to a wide range of trigonometric problems. Understanding these methods is crucial for success in trigonometry and related fields, as it allows for precise calculations and a deeper comprehension of trigonometric functions. By mastering the techniques outlined in this guide, you will be better prepared to tackle more complex trigonometric challenges and gain a stronger foundation in mathematics.

1. Understanding the Unit Circle and Trigonometric Functions

The unit circle is an indispensable tool in trigonometry. It’s a circle with a radius of 1, centered at the origin (0, 0) in the Cartesian coordinate system. The unit circle provides a visual representation of trigonometric functions for all angles, making it easier to understand their cyclical nature and values. Each point on the unit circle corresponds to an angle θ, measured counterclockwise from the positive x-axis. The coordinates of this point are given by (cos θ, sin θ), where the x-coordinate represents the cosine of the angle and the y-coordinate represents the sine of the angle. This direct link between angles and coordinates makes the unit circle an invaluable resource for evaluating trigonometric functions.

  • Sine (sin) corresponds to the y-coordinate of a point on the unit circle.
  • Cosine (cos) corresponds to the x-coordinate of a point on the unit circle.
  • Tangent (tan) is the ratio of sine to cosine (sin θ / cos θ).

Angles can be positive, measured counterclockwise, or negative, measured clockwise. A negative angle, such as -135°, indicates a rotation in the clockwise direction from the positive x-axis. By visualizing angles on the unit circle, we can quickly determine the signs and approximate values of their trigonometric functions. Understanding the relationship between angles and their coordinates on the unit circle is fundamental to mastering trigonometry. The unit circle not only simplifies the calculation of trigonometric values but also provides a clear visual aid for understanding the behavior of trigonometric functions across different quadrants. This foundational knowledge is essential for solving a wide range of trigonometric problems, from basic evaluations to more complex equation solving and graphing.

2. Identifying the Reference Angle

The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis. It is always a positive angle and helps simplify the calculation of trigonometric functions for angles outside the range of 0° to 90°. To find the reference angle, we need to consider the quadrant in which the angle lies. This determination is crucial as the reference angle serves as the foundation for calculating the trigonometric values of the original angle. By focusing on the acute angle formed with the x-axis, we can leverage our knowledge of trigonometric values in the first quadrant to find the values in other quadrants.

For negative angles, it's helpful to visualize the clockwise rotation on the unit circle. An angle of -135° means rotating 135° clockwise from the positive x-axis. This rotation places the terminal side of the angle in the third quadrant. In the third quadrant, both the x and y coordinates are negative, which will affect the signs of the trigonometric functions.

To calculate the reference angle (θ') for -135°, we can use the following logic:

Since -135° is in the third quadrant, we find the difference between the angle and the straight angle (-180°). However, because we are looking for a positive reference angle, we take the absolute value of the difference between -135° and -180°:

θ' = |-180° - (-135°)| = |-180° + 135°| = |-45°| = 45°

Therefore, the reference angle for -135° is 45°. This reference angle simplifies the process of finding the trigonometric values, as we can now relate the sine, cosine, and tangent of -135° to the known values of 45°. Understanding the concept of reference angles is key to efficiently calculating trigonometric values for any angle, regardless of its magnitude or sign. By systematically determining the reference angle, we can break down complex trigonometric problems into more manageable steps, ultimately leading to accurate solutions.

3. Determining the Sign of sin(-135°)

The sign of a trigonometric function in a particular quadrant is critical for determining the correct value. The unit circle provides a clear visual representation of these signs. In the first quadrant (0° to 90°), all trigonometric functions (sine, cosine, and tangent) are positive. However, in other quadrants, the signs vary depending on the function. Understanding these sign conventions is crucial for accurate trigonometric calculations. For instance, knowing that sine is negative in the third and fourth quadrants allows us to correctly apply the reference angle to find the actual value.

To determine the sign of sin(-135°), we again consider the quadrant in which the angle lies. As established earlier, -135° lies in the third quadrant. In the third quadrant:

  • Sine (sin) is negative because the y-coordinates are negative.
  • Cosine (cos) is negative because the x-coordinates are negative.
  • Tangent (tan) is positive because it is the ratio of sine to cosine (negative/negative = positive).

Since sine corresponds to the y-coordinate, sin(-135°) will be negative. This understanding of the sign is a critical step in finding the exact value. We know the reference angle is 45°, and we know that sin(-135°) is negative. Therefore, we can use the sine of the reference angle (sin 45°) and apply the appropriate negative sign. This systematic approach ensures that we account for both the magnitude and the direction of the angle when calculating its trigonometric functions. By mastering the sign conventions in each quadrant, we can confidently tackle a wide range of trigonometric problems and arrive at accurate solutions.

4. Evaluating sin(45°)

Now that we have identified the reference angle as 45°, we need to evaluate sin(45°). The value of sin(45°) is a fundamental trigonometric value that is often memorized or can be easily derived from special right triangles. These special triangles, particularly the 45-45-90 triangle, provide a straightforward way to calculate the sine, cosine, and tangent of key angles. Understanding these values is crucial for solving a wide range of trigonometric problems and forms the foundation for more advanced concepts.

A 45-45-90 triangle is an isosceles right triangle with two angles of 45° and one right angle (90°). The sides of this triangle are in the ratio 1:1:√2, where the two equal sides are of length 1, and the hypotenuse is of length √2. This ratio allows us to easily determine the trigonometric values for 45°.

Using the definition of sine as the ratio of the opposite side to the hypotenuse:

sin(45°) = Opposite / Hypotenuse = 1 / √2

To rationalize the denominator, we multiply both the numerator and the denominator by √2:

sin(45°) = (1 / √2) * (√2 / √2) = √2 / 2

Therefore, sin(45°) = √2 / 2. This value is positive, as expected for an angle in the first quadrant. This step is a critical component of finding sin(-135°), as we now have the magnitude of the sine value. By knowing the sine of the reference angle, we can easily determine the sine of the original angle by applying the correct sign, which we identified in the previous step. This systematic approach ensures accuracy and a deeper understanding of trigonometric functions.

5. Determining the Exact Value of sin(-135°)

We have now gathered all the necessary pieces to determine the exact value of sin(-135°). We have identified the reference angle, determined the sign of sin(-135°), and evaluated sin(45°). The final step is to combine these elements to find the solution. This process demonstrates the systematic approach to solving trigonometric problems, where each step builds upon the previous one to arrive at the correct answer.

We know that:

  • The reference angle for -135° is 45°.
  • sin(-135°) is negative (since -135° lies in the third quadrant).
  • sin(45°) = √2 / 2

Therefore, to find the exact value of sin(-135°), we take the value of sin(45°) and apply the negative sign:

sin(-135°) = -sin(45°) = -√2 / 2

Thus, the exact value of sin(-135°) is -√2 / 2. This result provides a precise value for the sine of the angle, avoiding any approximations. The systematic method we followed, including finding the reference angle, determining the sign, and using known trigonometric values, can be applied to calculate the exact trigonometric values of various angles. By mastering this approach, you can confidently solve trigonometric problems and gain a deeper understanding of trigonometric functions. The ability to determine exact values is a valuable skill in mathematics and related fields, allowing for precise calculations and a solid foundation for further study.

Conclusion

In conclusion, determining the exact value of sin(-135°) involves a series of logical steps grounded in the principles of trigonometry and the unit circle. By systematically identifying the reference angle, understanding the sign conventions in different quadrants, and leveraging known trigonometric values, we can accurately calculate the sine of any angle. This method not only provides the correct answer but also enhances our comprehension of trigonometric functions and their behavior. Mastering these techniques is essential for success in mathematics and related fields, as it allows for precise calculations and a deeper appreciation of trigonometric concepts. The ability to break down complex problems into manageable steps and apply fundamental principles is a hallmark of strong mathematical understanding. By practicing these methods and applying them to a variety of problems, you can build confidence in your trigonometric skills and achieve mastery in this critical area of mathematics.