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5 votes
5 votes
Find the general solution and find the specific solutions over the interval [0,4pi]

cos(alpha + pi/3) = (sqrt(3))/2

User Barry Colebank Jr
by
2.8k points

1 Answer

4 votes
4 votes

Answer:

Isolate the angle 2x, by following the reverse "order of operations".

Explanation:

Explanation:

Step 1: Add 1 to both sides:

2

cos

2

(

2

x

)

=

1

Step 2: Divide both sides by 2:

cos

2

(

2

x

)

=

1

2

Step 3: Take the square root of both sides:

cos

(

2

x

)

=

2

2

or

cos

(

2

x

)

=

2

2

(don't forget the positive and negative solutions!)

Step 4: Use inverse of cosine to find the angles:

2

x

=

cos

1

(

2

2

)

or

2

x

=

cos

1

(

2

2

)

Step 5: Find angles that work:

2

x

=

π

4

or

2

x

=

7

π

4

or

2

x

=

3

π

4

or

2

x

=

5

π

4

Step 6: Solve for x:

x

=

π

8

,

7

π

8

,

3

π

8

,

5

π

8

or .785, 5.5, 2.36, 3.93

(decimal approximations are seen on the graph below)

User Alex Tingle
by
2.6k points