Problem 1 - Contemporary Precalculus

Question

Find all solutions of the equation. $$\sin x=.465$$

Step-by-Step Solution

Answer

Question: Find all solutions of the equation \(\sin x = 0.465\). Answer: The general solutions of the equation \(\sin x = 0.465\) are \(x = 0.4794 + 2\pi k\) or \(x = 2.6622 + 2\pi k\) for all integers \(k\).

Step 1: Calculate the principal angle

The principal angle is the angle that is in the interval of \([0, 2\pi]\) and satisfies the equation \(\sin x = 0.465\). Use a calculator to find the inverse sine (also known as arcsin) to find the angle: $$x = \arcsin(0.465) = 0.4794 \text{ radians}$$

Step 2: Determine the values in the second quadrant

Since the sine function is positive in both the first and second quadrants, there is another angle in the second quadrant that gives the same sine value. To find that angle, subtract the principal angle value obtained in Step 1 from \(\pi\): $$x_2 = \pi - 0.4794 = 2.6622 \text{ radians}$$

Step 3: Determine the general solutions

Now that we have the principal angle and the angle in the second quadrant, we need to find all general solutions to the equation. Since the sine function has a period of \(2\pi\), we can express the general solutions as: $$x = 0.4794 + 2\pi k \text{ or } x = 2.6622 + 2\pi k$$ where \(k\) is an integer. So, the general solutions of the equation \(\sin x = 0.465\) are \(x = 0.4794 + 2\pi k\) or \(x = 2.6622 + 2\pi k\) for all integers \(k\).

Key Concepts

Sine Function

The sine function, often written as \(\sin(x)\), is a fundamental concept in trigonometry. It relates the angle \(x\) in a right triangle to the ratio of the length of the opposite side to the hypotenuse. Sine functions are periodic, meaning they repeat values in a predictable manner. The standard sine curve

• has a period of \(2\pi\), which is the length of one complete cycle
• oscillates between -1 and 1
• reaches its maximum at \(\frac{\pi}{2}\) and minimum at \(\frac{3\pi}{2}\)
• is symmetrical around the origin

Understanding the sine function is essential when solving trigonometric equations, as it helps predict and find various angles that produce the same sine values. The central role of the sine curve in trigonometry helps explain certain periodic phenomena in physics and engineering.

Principal Angle

The principal angle is a crucial concept when dealing with trigonometric equations. When asked to solve for angles, the principal angle is the smallest angle that satisfies the equation in the standard interval \[ [0, 2\pi] \] For example, to find the principal angle for \(\sin(x) = 0.465\), you use the inverse sine function, \(\arcsin(0.465)\), to determine the angle \(x = 0.4794\) radians.
Since the sine function is positive in both the first and second quadrants, there may also be another possible angle in the second quadrant which needs to be considered.
Finding the principal angle determines one of the possible solutions for a trigonometric equation, and in many cases, it represents the "most natural" solution to an equation.

General Solutions

General solutions in trigonometric equations allow us to find all possible angles that satisfy the given equation. Because trigonometric functions are periodic, there's often more than one angle that produces the same outcome. For example, when finding solutions for \(\sin(x) = 0.465\), we not only find the angles within a single cycle but also extend to include angles from all other cycles.The solutions for sine functions include both the principal angle and its counterpart in the second quadrant (since \(\sin\) is positive there as well). This leads to expressions for an infinite number of solutions, represented as:

• \( x = 0.4794 + 2\pi k \)
• \( x = 2.6622 + 2\pi k \)

where \( k \) is any integer. By using these expressions, we accommodate the infinite nature of sine wave repetitions, ensuring all angles that produce a sine value of 0.465 are considered.

Radians

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians use the circle's circumference as the unit of measurement. In this system:

• One full rotation (circle) is \(2\pi\) radians
• \(\pi\) radians equal 180 degrees
• 1 radian is about 57.3 degrees

Radians are particularly useful in mathematics because they simplify many formulas, especially those involving trigonometry.
For example, when working with the sine function, angles are typically expressed in radians since it aligns well with the periodic nature of trigonometric functions.In solving \(\sin(x) = 0.465\), outputs are often given in radians, making it important to understand this measurement to correctly interpret angles and solutions.