Problem 26 - Algebra and Trigonometry

Question

\(25-38\) . Find all solutions of the given equation. $$ \sin \theta+1=0 $$

Step-by-Step Solution

Answer

The solution is \( \theta = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer.

Step 1: Understand the Equation

The given equation is \( \sin \theta + 1 = 0 \). This can be rewritten by isolating \( \sin \theta \): \( \sin \theta = -1 \).

Step 2: Identify the Range of Sine Function

Recall that the sine function, \( \sin \theta \), can take values between -1 and 1, inclusive. This means that \( \sin \theta = -1 \) is within this range and possible.

Step 3: Determine the Angle(s) for Sine

The sine function equals -1 at specific angles. Recall that \( \sin \theta = -1 \) at \( \theta = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer, because this represents a full cycle of the sine wave.

Key Concepts

Sine Function

The sine function, written as \( \sin \theta \), is a fundamental concept in trigonometry. It relates to the ratio of the length of the opposite side of an angle to the hypotenuse in a right-angled triangle. The sine function is one of the primary functions used to describe wave patterns and oscillations in mathematics and physics. It operates on an angle \( \theta \), which can be measured in degrees or radians.
The sine function is periodic with a cycle of \( 2\pi \) radians, meaning that its values repeat every \( 2\pi \). This property is crucial in solving trigonometric equations, as it allows for the prediction of multiple solutions for any given sine value.

Angle Solutions

When solving equations involving trigonometric functions like sine, finding angle solutions is the key task. For each sine value, there can be several angles that satisfy the equation, because of the periodic nature of the sine function.
The fundamental solution emerges directly from where the sine value aligns with typical angles on the unit circle. For example, if \( \sin \theta = -1 \), the angle solution is \( \theta = \frac{3\pi}{2} \), since sine reaches -1 precisely at this angle.

• However, to find all possible solutions, consider the periodic nature: repeat the cycle by adding multiples of \( 2\pi \).

This means that the full set of solutions can be described as \( \theta = \frac{3\pi}{2} + 2k\pi \), where \( k \) is any integer. This expression captures all angles that will satisfy the original sine equation.

Range of Sine Function

Understanding the range of the sine function is important in trigonometry, as it defines the set of possible output values for \( \sin \theta \). This range is from -1 to 1, inclusive.
This means that for any value of \( \theta \), \( \sin \theta \) cannot exceed 1 or be less than -1. The sine function reaches its maximum value of 1 at \( \theta = \frac{\pi}{2} + 2k\pi \) and minimum value of -1 at \( \theta = \frac{3\pi}{2} + 2k\pi \).

• So, whenever you encounter an equation like \( \sin \theta = -1 \), you know it's a legitimate scenario since -1 is within the function's range.

This understanding assists in solving trigonometric equations, as it frames what possible solutions might look like.

Unit Circle

The unit circle is an essential tool in trigonometry, particularly helpful in understanding and visualizing the sine function. It is a circle with a radius of 1 unit centered at the origin of a coordinate plane.
In the context of the unit circle, angles are measured from the positive x-axis, counterclockwise around the circle. The sine of any angle \( \theta \) is the y-coordinate of the corresponding point on the unit circle. This relationship helps in finding sine values at common angles and solving trigonometric equations.

• For example, when \( \sin \theta = -1 \), this corresponds to the point directly downward on the circle, linking to the angle \( \theta = \frac{3\pi}{2} \).

By visualizing these angles on the unit circle, you can easily determine their periodic repetitions, which give rise to additional solutions for angles.