Identifying Negative Trigonometric Values Evaluating Cos(3.78) Ctg(2.91) And Tg(4.45)
In the realm of trigonometry, understanding the signs of trigonometric functions in different quadrants is crucial. Trigonometric functions like cosine (cos), cotangent (ctg), and tangent (tg) exhibit varying signs depending on the angle's quadrant. In this article, we delve into evaluating these functions for given angles and determining whether their values are negative. This involves understanding the unit circle, the definitions of trigonometric functions, and their behavior in different quadrants. Trigonometric functions are at the heart of many mathematical and real-world applications, from physics to engineering, making a solid grasp of their properties essential. This discussion focuses on how to identify the signs of trigonometric functions for specific angles. In this exploration, we will analyze the given angles and use our knowledge of trigonometric functions to determine which of them result in negative values.
The foundation for understanding the signs of trigonometric functions lies in the unit circle. 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. A point on the unit circle is defined by its coordinates (x, y), where x represents the cosine of the angle and y represents the sine of the angle. The tangent of the angle is given by the ratio of sine to cosine (y/x), and the cotangent is the reciprocal of the tangent (x/y). The unit circle provides a visual representation of how these functions change signs in different quadrants. In the first quadrant (0 to π/2), all trigonometric functions are positive. In the second quadrant (π/2 to π), sine is positive, while cosine and tangent are negative. In the third quadrant (π to 3π/2), tangent is positive, while sine and cosine are negative. Finally, in the fourth quadrant (3π/2 to 2π), cosine is positive, while sine and tangent are negative. Analyzing trigonometric functions involves using the unit circle to determine the sign of each function based on the angle's quadrant. This requires understanding how sine, cosine, tangent, and cotangent behave in each quadrant. The unit circle, with its quadrants, acts as a fundamental tool in determining the signs of trigonometric functions for different angles.
To determine which of the given numbers are negative, we need to evaluate the trigonometric functions for the specified angles and consider their quadrants. The given numbers are:
- cos(3.78)
- ctg(2.91)
- tg(4.45)
We will analyze each expression individually, converting the angles from radians to a more intuitive quadrant representation.
1) Evaluating cos(3.78)
The angle 3.78 radians lies in the third quadrant because π < 3.78 < 3π/2 (approximately 3.14 < 3.78 < 4.71). In the third quadrant, cosine is negative since the x-coordinate is negative. Therefore, cos(3.78) is negative. The cosine function is negative in the second and third quadrants. To accurately determine the sign, we need to locate the angle within the unit circle. Since 3.78 radians falls into the third quadrant, where x-coordinates are negative, cos(3.78) yields a negative value. Understanding the properties of the cosine function across different quadrants helps in this evaluation.
2) Evaluating ctg(2.91)
The angle 2.91 radians lies in the second quadrant because π/2 < 2.91 < π (approximately 1.57 < 2.91 < 3.14). Cotangent is the reciprocal of tangent, and tangent is negative in the second quadrant (since sine is positive and cosine is negative). Therefore, cotangent is also negative in the second quadrant. Thus, ctg(2.91) is negative. The cotangent function is defined as the ratio of cosine to sine. In the second quadrant, cosine is negative and sine is positive, making the cotangent negative. Thus, by locating 2.91 radians within the second quadrant, we confirm that ctg(2.91) is a negative value. This illustrates how understanding quadrant rules aids in evaluating the sign of the cotangent function.
3) Evaluating tg(4.45)
The angle 4.45 radians lies in the third quadrant because π < 4.45 < 3π/2 (approximately 3.14 < 4.45 < 4.71). In the third quadrant, tangent is positive because both sine and cosine are negative, and a negative divided by a negative is positive. Therefore, tg(4.45) is positive. The tangent function is positive in the first and third quadrants. By determining that 4.45 radians falls within the third quadrant, where both sine and cosine are negative, we find that tg(4.45) is positive. This determination is based on the understanding of how the tangent function's sign changes across the unit circle.
Based on our analysis:
- cos(3.78) is negative.
- ctg(2.91) is negative.
- tg(4.45) is positive.
Therefore, the negative numbers are 1 and 2. The correct answer is A) 1; 2. In summary, evaluating the signs of trigonometric functions requires a strong understanding of the unit circle and the properties of sine, cosine, tangent, and cotangent in each quadrant. This methodical approach helps in accurately determining whether trigonometric functions result in positive or negative values for given angles. Understanding the quadrants and their influence on trigonometric function signs is essential for solving such problems accurately and efficiently.
Therefore, the final answer is A) 1; 2.
