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Chapter 7: Problem 8

Solve, finding all solutions. Express the solutions in both radians and degrees. $$\sin x=\frac{\sqrt{3}}{2}$$

Short Answer

Expert verified
The solutions are \ (\ x = \frac{\pi}{3}+ 2k\pi \) and \( x = \frac{2\pi}{3} + 2k\pi \) in radians; \( x = 60^\circ + 360^\circ k \) and \( x = 120^\circ + 360^\circ k \) in degrees.

Step by step solution

01

- Identify Known sine Values

Recall the commonly known sine values. We know that \(\begin{aligned}\sin \frac{\pi}{3} &= \frac{\sqrt{3}}{2} \sin \left(\pi - \frac{\pi}{3}\right) &= \frac{\sqrt{3}}{2} \end{aligned}\)
02

- Write General Solutions in Radians

The general solutions for sine, taking into account its periodic nature, are given by the equations \(x = \frac{\pi}{3} + 2k\pi\setminus\text{and} x = \frac{2\pi}{3} + 2k\pi \) where \(k\) is any integer.
03

- Convert to Degrees

Convert the radian measures to degrees using the formula \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \). This yields the solutions \( x = 60^\circ + 360^\circ k \) and \( x = 120^\circ + 360^\circ k \), where \(k\) is any integer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sine function
The sine function, denoted as \(\backslashsin x\), is one of the fundamental trigonometric functions. It represents the y-coordinate of a point on the unit circle corresponding to a given angle measured from the positive x-axis.

The sine function oscillates between -1 and 1. Its value varies smoothly and periodically as the angle changes.

Key properties of the sine function include:
  • Amplitude: The maximum value (\backslashsin x can reach is 1, and the minimum value is -1.
  • Periodicity: The sine function has a period of \(2\backslashpi\), meaning \( \backslashsin(x+2\backslashpi) = \backslashsin x\). This repeating nature is crucial for solving equations involving sine.
  • Odd Function: The sine function is odd, meaning \( \backslashsin(-x) = -\backslashsin(x)\).
  • Symmetry: It is symmetric about the origin.
radian measure
Radians are the standard unit of angular measure used in many areas of mathematics. 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.

To relate radians to degrees, remember that a full circle is \(2\backslashpi\) radians, which is equivalent to 360 degrees. Therefore, to convert radians to degrees, use the conversion:
\[ \text{Degrees} = \text{Radians} \times \frac{180}{\backslashpi} \]

For example, \(\frac{\backslashpi}{3}\) radians can be converted to degrees as follows:
\[ \frac{\backslashpi}{3} \times \frac{180}{\backslashpi} = 60^\text{circ} \]
Understanding radian measure is vital when working with trigonometric functions and their periodic properties.
degree measure
Degrees are another unit of angular measurement, commonly used in everyday applications. A full circle is divided into 360 degrees.

To convert degrees to radians, use the formula:
\[ \text{Radians} = \text{Degrees} \times \frac{\backslashpi}{180} \]

For instance, converting 120 degrees to radians involves:
\[ 120^\text{circ} \times \frac{\backslashpi}{180} = \frac{2\backslashpi}{3} \]
Degrees provide a more intuitive grasp for angles, especially when dealing with geometrical and real-world problems.
periodicity of trigonometric functions
Trigonometric functions are periodic, meaning they repeat their values at regular intervals. For the sine function, this interval is \(2\backslashpi\).

The periodic nature of trigonometric functions is helpful in solving equations because it allows for general solutions. For the sine function, given a specific value such as \( \backslashsin x = \frac{\backslashsqrt{3}}{2} \), the general solutions can be expressed as:
  • \( x = \frac{\backslashpi}{3} + 2k\backslashpi \)
  • \( x = \frac{2\backslashpi}{3} + 2k\backslashpi \)
where \( k \) is any integer.
The periodicity of \(2\backslashpi\) allows for these infinite solutions by adding integer multiples of \(2\backslashpi\). This concept is crucial for finding all possible solutions to trigonometric equations.

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Most popular questions from this chapter

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { linear speed } & \text { congruent } \\ \text { angular speed } & \text { circular } \\ \text { angle of elevation } & \text { periodic } \\ \text { angle of depression } & \text { period } \\ \text { complementary } & \text { amplitude } \\ \text { supplementary } & \text { quadrantal } \\ \text { similar } & \text { radian measure }\end{array}$$ The _____________ of a rotation is the ratio of the distance \(s\) traveled by a point at a radius \(r\) from the center of rotation to the length of the radius \(r .\)

SOLVE. $$\cos ^{-1} x=\cos ^{-1} \frac{3}{5}-\sin ^{-1} \frac{4}{5}$$

Use the given substitution to express the given radical expression as a trigonometric function without radicals. Assume that \(a>0\) and \(0<\theta<\pi / 2 .\) Then find expressions for the indicated trigonometric functions. Let \(x=3 \sec \theta\) in \(\sqrt{x^{2}-9}\). Then find \(\sin \theta\) and \(\cos \theta\)

First write each of the following as a trigonometric function of a single angle. Then evaluate. $$\frac{\tan 35^{\circ}-\tan 12^{\circ}}{1+\tan 35^{\circ} \tan 12^{\circ}}$$

Angles Between Lines. One of the identities gives an easy way to find an angle formed by two lines. Consider two lines with equations \(l_{1}: y=m_{1} x+b_{1}\) and \(l_{2}: y=m_{2} x+b_{2}\) (GRAPH CANNOT COPY) The slopes \(m_{1}\) and \(m_{2}\) are the tangents of the angles \(\theta_{1}\) and \(\theta_{2}\) that the lines form with the positive direction of the \(x\) -axis. Thus we have \(m_{1}=\tan \theta_{1}\) and \(m_{2}=\tan \theta_{2} .\) To find the measure of \(\theta_{2}-\theta_{1},\) or \(\phi,\) we proceed as follows: This formula also holds when the lines are taken in the reverse order. When \(\phi\) is acute, tan \(\phi\) will be positive. When \(\phi\) is obtuse, tan \(\phi\) will be negative. Find the measure of the angle from \(l_{1}\) to \(l_{2}\) $$\begin{aligned} &l_{1}: 2 x=3-2 y\\\ &l_{2}: x+y=5 \end{aligned}$$
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