Question
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Solve the equation. Give a general formula for all the solutions. List six solutions.
Identify the general formula for all the solutions to based on the smaller angle,
is an integer
(Simplify your answer. Use angle measures greater than or equal to 0 and less than . Type an exact answer, using as needed. Use integers or fractions for any numbers in the expression. Type
an expression using k as the variable.)

Ask by Joseph Wade. in the United States
Jan 20,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The general solution for is:
Six specific solutions within are:

Solution

To solve the equation:
we seek all angles that satisfy this equation. The sine function equals at specific standard angles within the interval .

Step 1: Identify the Reference Angles

The value corresponds to the sine of radians (or 60 degrees). Since the sine function is positive in both the first and second quadrants, the two primary solutions within one period are:

Step 2: Generalize the Solutions

The sine function has a period of , meaning the solutions repeat every radians. Therefore, the general solutions can be expressed as:

Step 3: List Six Specific Solutions

To provide six specific solutions within the interval , we can consider two full periods ( to ):
  1. For :
  2. For :
  3. For :

Summary

General Solution:
Six Specific Solutions:

Answered by UpStudy AI and reviewed by a Professional Tutor

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Explain
Simplify this solution

Bonus Knowledge

To solve the equation , we first identify the angles where the sine function takes this value. The two primary angles in the range are and . The general formula for all solutions can be expressed as:
Now, let’s list six specific solutions by choosing integer values for :
  1. For : and
  2. For : and
  3. For : (but we can keep it positive, so we can use equivalent positive angle: ) and (equivalent angle: )
The six specific solutions are:
  1. (this one repeats)
So, adjusting for distinct values we get unique solutions of:
  1. (as the next valid solution)
In summary, the unique solutions include angles that showcase the periodic nature of sine!

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