Question
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.)
(Simplify your answer. Use angle measures greater than or equal to 0 and less than
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
):
-
For
: -
-
For
: -
-
For
: -
Summary
General Solution:
Six Specific Solutions:
Answered by UpStudy AI and reviewed by a Professional Tutor
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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
:
- For
: and - For
: and - For
: (but we can keep it positive, so we can use equivalent positive angle: ) and (equivalent angle: )
The six specific solutions are:
-
-
-
-
-
-
(this one repeats)
So, adjusting for distinct values we get unique solutions of:
-
-
-
-
-
-
(as the next valid solution)
In summary, the unique solutions include angles that showcase the periodic nature of sine!