Problem 44
Question
In Exercises 39-44, solve the multiple-angle equation. \( \sin \dfrac{x}{2} = -\dfrac{\sqrt{3}}{2} \)
Step-by-Step Solution
Verified Answer
The solutions for the equation are \(x = 8\pi/3\) and \(x = 10\pi/3\).
1Step 1: Identify the angle
In a unit circle, the sine function equals to \(-\sqrt{3}/2\) at two places: 240 degrees or \(4\pi/3\) radians and 300 degrees or \(5\pi/3\) radians. Therefore, \(x/2 = 4\pi/3\) and \(x/2 = 5\pi/3\).
2Step 2: Solve for x
Solving the equations from the first step for x, we get \(x = 2 * (4\pi / 3) = 8\pi/3\) and \(x = 2 * (5\pi / 3) = 10\pi/3\).
Key Concepts
Trigonometric FunctionsUnit CircleRadians
Trigonometric Functions
Trigonometric functions are fundamental in the study of periodic phenomena and geometrical relationships involving triangles. They are integral to solving equations involving angles, such as the multiple-angle equation in the given exercise.
Among the primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which relate an angle of a right-angled triangle to the ratios of two sides of the triangle. In the context of the unit circle, these functions can also be seen as the coordinates of a point on the circle's circumference, where the angle is measured from the positive x-axis.
For example, the equation from the exercise, \( \text{sin} \dfrac{x}{2} = -\dfrac{\sqrt{3}}{2} \), indicates that we are looking for angles where the sine function attains the value of \( -\sqrt{3}/2 \). This is found by assessing the y-coordinate of a point on the unit circle that makes an angle of \( \dfrac{x}{2} \) with the positive x-axis.
Among the primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), which relate an angle of a right-angled triangle to the ratios of two sides of the triangle. In the context of the unit circle, these functions can also be seen as the coordinates of a point on the circle's circumference, where the angle is measured from the positive x-axis.
For example, the equation from the exercise, \( \text{sin} \dfrac{x}{2} = -\dfrac{\sqrt{3}}{2} \), indicates that we are looking for angles where the sine function attains the value of \( -\sqrt{3}/2 \). This is found by assessing the y-coordinate of a point on the unit circle that makes an angle of \( \dfrac{x}{2} \) with the positive x-axis.
Unit Circle
The unit circle is a vital concept in trigonometry, representing a circle with a radius of 1 unit centered at the origin of a coordinate plane. Every point on the unit circle has coordinates that correspond to the cosine and sine of the angle formed by the line connecting the point to the circle's center and the positive x-axis.
This circle is particularly useful when working with trigonometric functions over various quadrants. Since the radius of the unit circle is always 1, the x-coordinate of any point on the circle is equal to the cosine of the angle, and the y-coordinate is equal to the sine of the angle.
In our exercise, the negative sine value suggests that the point corresponding to the angle \( \dfrac{x}{2} \) lies below the x-axis, which happens in the third and fourth quadrants of the circle. This is key in identifying the correct angles in the solution.
This circle is particularly useful when working with trigonometric functions over various quadrants. Since the radius of the unit circle is always 1, the x-coordinate of any point on the circle is equal to the cosine of the angle, and the y-coordinate is equal to the sine of the angle.
In our exercise, the negative sine value suggests that the point corresponding to the angle \( \dfrac{x}{2} \) lies below the x-axis, which happens in the third and fourth quadrants of the circle. This is key in identifying the correct angles in the solution.
Radians
Radians are a unit of angular measure used in mathematics to express angles in a way that is particularly convenient when dealing with trigonometric functions and circular motion. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.
There are \( 2\pi \) radians in a full circle, which is equivalent to 360 degrees. Therefore, to convert degrees to radians, you multiply by \( \frac{\pi}{180} \) and for radians to degrees, you multiply by \( \frac{180}{\pi} \).
In our solved exercise, when the angle measures such as \( \frac{4\pi}{3} \) and \( \frac{5\pi}{3} \) are mentioned, they are using radians to specify the location on the unit circle where the sine value is \( -\sqrt{3}/2 \). The use of radians can sometimes simplify mathematical expressions and calculations, which is why they are preferred in higher mathematics.
There are \( 2\pi \) radians in a full circle, which is equivalent to 360 degrees. Therefore, to convert degrees to radians, you multiply by \( \frac{\pi}{180} \) and for radians to degrees, you multiply by \( \frac{180}{\pi} \).
In our solved exercise, when the angle measures such as \( \frac{4\pi}{3} \) and \( \frac{5\pi}{3} \) are mentioned, they are using radians to specify the location on the unit circle where the sine value is \( -\sqrt{3}/2 \). The use of radians can sometimes simplify mathematical expressions and calculations, which is why they are preferred in higher mathematics.
Other exercises in this chapter
Problem 44
Exercises 43-52, use the power-reducing formulas to rewrite the expression in terms of the first power of the cosine. \( \sin^4 2x \)
View solution Problem 44
In Exercises \(43-50\) , find the exact value of the trigonometric function given that \(\sin u=\frac{5}{13}\) and \(\cos v=-\frac{3}{5} .\) Both \(u\) and \(v\
View solution Problem 44
In Exercises 9-50, verify the identity \( \sec^2 y - \cot^2 \left(\dfrac{\pi}{2} - y \right) = 1 \)
View solution Problem 44
In Exercises 37 - 58, use the fundamental identities to simplify the expression. There is more than one correct form of each answer. \( \dfrac{\csc \theta}{\sec
View solution