Do I Half to (Part I)? Using Half Angle Formulas to find Exact Values
Professor Lively’s Video Channel・2 minutes read
The angle alpha has a tangent of \( \frac{4}{3} \) in the third quadrant, leading to sine and cosine values of \( \frac{-4}{5} \) and \( \frac{-3}{5} \), respectively. Consequently, the sine, cosine, and tangent of \( \frac{\alpha}{2} \) are \( \frac{2\sqrt{5}}{5} \), \( -\frac{1}{\sqrt{5}} \), and \( -2 \).
Insights
- The tangent of angle alpha, given as \( \frac{4}{3} \), leads to the determination of its sine and cosine values using the coordinates in the third quadrant, resulting in sine being \( \frac{-4}{5} \) and cosine being \( \frac{-3}{5} \). This sets the foundation for calculating the sine and cosine of \( \frac{\alpha}{2} \), which falls in the second quadrant where sine is positive.
- To find the sine and cosine of \( \frac{\alpha}{2} \), specific formulas are applied: the sine is calculated as \( \sin\left(\frac{\alpha}{2}\right) = \frac{2\sqrt{5}}{5} \) and the cosine as \( \cos\left(\frac{\alpha}{2}\right) = -\frac{1}{\sqrt{5}} \). This leads to the tangent of \( \frac{\alpha}{2} \) being \( -2 \), illustrating the relationship between the angles and their trigonometric functions across different quadrants.
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Recent questions
What is the sine of an angle?
The sine of an angle is a fundamental trigonometric function that represents the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. It is denoted as sin(θ), where θ is the angle in question. The sine function is periodic, with a range of values between -1 and 1, and it plays a crucial role in various applications, including physics, engineering, and computer graphics. In the unit circle, the sine of an angle corresponds to the y-coordinate of the point on the circle at that angle, providing a geometric interpretation of the function.
How do I calculate cosine values?
To calculate cosine values, you can use the cosine function, which is another fundamental trigonometric function. The cosine of an angle θ, denoted as cos(θ), is defined as the ratio of the length of the adjacent side to the hypotenuse in a right triangle. For angles measured in radians or degrees, you can use a scientific calculator or trigonometric tables to find the cosine value directly. Additionally, the cosine function is periodic and has a range of values from -1 to 1. In the context of the unit circle, the cosine of an angle corresponds to the x-coordinate of the point on the circle at that angle, allowing for a visual understanding of the function.
What is the tangent of an angle?
The tangent of an angle is a trigonometric function that represents the ratio of the sine of the angle to the cosine of the angle. It is denoted as tan(θ) and can also be understood as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. The tangent function is periodic and has a range of all real numbers, with vertical asymptotes where the cosine of the angle is zero. In practical applications, the tangent function is used in various fields such as physics, engineering, and navigation, making it essential for solving problems involving angles and distances.
What is the unit circle in trigonometry?
The unit circle is a fundamental concept in trigonometry that represents a circle with a radius of one centered at the origin of a coordinate plane. It is used to define the sine, cosine, and tangent functions for all angles, not just those in right triangles. Each point on the unit circle corresponds to an angle measured from the positive x-axis, and the coordinates of these points provide the values of the cosine and sine for that angle. The unit circle is particularly useful for understanding the periodic nature of trigonometric functions and for solving trigonometric equations, as it allows for a visual representation of angles and their corresponding sine and cosine values.
How do I find the sine and cosine of an angle?
To find the sine and cosine of an angle, you can use the definitions of these trigonometric functions based on a right triangle or the unit circle. For a right triangle, the sine of an angle is the ratio of the length of the opposite side to the hypotenuse, while the cosine is the ratio of the length of the adjacent side to the hypotenuse. Alternatively, if you are working with the unit circle, you can determine the sine and cosine of an angle by identifying the coordinates of the point on the circle corresponding to that angle. For angles measured in degrees or radians, you can also use a scientific calculator or trigonometric tables to obtain the sine and cosine values directly.
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Summary
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Trigonometric Values for Half Angle Alpha
- The tangent of angle alpha is given as \( \frac{4}{3} \), and alpha is located in the third quadrant, specifically between \( \pi \) and \( \frac{3\pi}{2} \). To find the sine, cosine, and tangent of \( \frac{\alpha}{2} \), we first need to determine the sine and cosine of alpha using the relationship between the tangent and the coordinates in the third quadrant, where \( x = -3 \) and \( y = -4 \).
- Using the Pythagorean theorem, \( x^2 + y^2 = r^2 \), we calculate \( r \) as follows: \( (-3)^2 + (-4)^2 = 25 \), leading to \( r = 5 \). Consequently, the sine of alpha is \( \frac{y}{r} = \frac{-4}{5} \) and the cosine of alpha is \( \frac{x}{r} = \frac{-3}{5} \).
- Since alpha is in the third quadrant, \( \frac{\alpha}{2} \) will be in the second quadrant, where sine is positive and cosine and tangent are negative. The range for \( \frac{\alpha}{2} \) is determined by halving the bounds of alpha, resulting in \( \frac{\pi}{2} < \frac{\alpha}{2} < \frac{3\pi}{4} \).
- To find the sine of \( \frac{\alpha}{2} \), we use the formula \( \sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \cos(\alpha)}{2}} \). Substituting \( \cos(\alpha) = -\frac{3}{5} \), we have \( \sin\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - (-\frac{3}{5})}{2}} = \sqrt{\frac{8/5}{2}} = \frac{2\sqrt{5}}{5} \).
- For the cosine of \( \frac{\alpha}{2} \), we apply the formula \( \cos\left(\frac{\alpha}{2}\right) = -\sqrt{\frac{1 + \cos(\alpha)}{2}} \). Substituting \( \cos(\alpha) = -\frac{3}{5} \), we find \( \cos\left(\frac{\alpha}{2}\right) = -\sqrt{\frac{1 - \frac{3}{5}}{2}} = -\sqrt{\frac{2/5}{2}} = -\frac{1}{\sqrt{5}} \). The tangent of \( \frac{\alpha}{2} \) is then calculated as \( \tan\left(\frac{\alpha}{2}\right) = \frac{\sin\left(\frac{\alpha}{2}\right)}{\cos\left(\frac{\alpha}{2}\right)} = -2 \).
