Solve the equation. Give a general formula for all the solutions. List six solutions.
[tex]\sin \theta=\frac{\sqrt{2}}{2}[/tex]
Write the general formula for all the solutions to [tex]\sin \theta=\frac{\sqrt{2}}{2}[/tex] based on the larger angle.
[tex]\theta=\frac{3 \pi}{4}+2 k \pi[/tex]
(Simplify your answer. Use angle measures greater than or equal to 0 and less than [tex]2 \pi[/tex]. Type for any numbers in the expression. Type an expression using [tex]k[/tex] as the variable.)
List the first six solutions that are greater than or equal to 0.
[tex]\theta=[/tex]
(Simplify your answer. Type an exact answer, using [tex]\pi[/tex] as needed. Use integers or fractions for
Answer (1)
Find the principal solutions for θ : 4 π and 4 3 π .
Express the general solution: θ = 4 π + 2 kπ or θ = 4 3 π + 2 kπ , where k is an integer.
Use the given general formula θ = 4 3 π + 2 kπ and the other general formula to find the first six non-negative solutions.
The first six solutions are: 4 π , 4 3 π , 4 9 π , 4 11 π , 4 17 π , 4 19 π .
Explanation
Problem Analysis We are given the equation sin θ = 2 2 and asked to find a general formula for all solutions and list six solutions.
Finding Principal Solutions First, we find the principal solutions for θ in the interval [ 0 , 2 π ) . The two angles whose sine is 2 2 are 4 π and 4 3 π .
General Solution The general solution can be expressed as θ = 4 π + 2 kπ or θ = 4 3 π + 2 kπ , where k is an integer.
Given General Formula We are given the general formula based on the larger angle: θ = 4 3 π + 2 kπ .
Finding Six Non-Negative Solutions Now, we need to find the first six non-negative solutions. We will use both general formulas: θ = 4 π + 2 kπ and θ = 4 3 π + 2 kπ .
Solutions from First Formula For θ = 4 π + 2 kπ :
When k = 0 , θ = 4 π
When k = 1 , θ = 4 π + 2 π = 4 π + 4 8 π = 4 9 π
When k = 2 , θ = 4 π + 4 π = 4 π + 4 16 π = 4 17 π
Solutions from Second Formula For θ = 4 3 π + 2 kπ :
When k = 0 , θ = 4 3 π
When k = 1 , θ = 4 3 π + 2 π = 4 3 π + 4 8 π = 4 11 π
When k = 2 , θ = 4 3 π + 4 π = 4 3 π + 4 16 π = 4 19 π
Combining Solutions Combining these, the first six non-negative solutions are 4 π , 4 3 π , 4 9 π , 4 11 π , 4 17 π , 4 19 π .
Final Answer Therefore, the first six solutions are θ = 4 π , 4 3 π , 4 9 π , 4 11 π , 4 17 π , 4 19 π .
Examples
Understanding trigonometric equations like sin ( θ ) = 2 2 is crucial in many real-world applications. For instance, in physics, it helps describe the motion of a pendulum or the behavior of alternating current in electrical circuits. Imagine designing a swing set where the angle of the swing at certain points needs to be precisely determined for safety and performance. Solving such equations allows engineers to predict and control these angles accurately, ensuring a smooth and safe swinging experience. Similarly, in music, sine waves are used to model sound waves, and understanding their properties helps in creating harmonious sounds and designing acoustic spaces.
