Asked by Tiger
                csc(6a+ð/8)=sec(2a-ð/8)
Find values of A.
            
        Find values of A.
Answers
                    Answered by
            MathMate
            
    Since the original question was not posted in Western encoding, the strange character would reasonably be interpreted as π.  So the question would be:
<i>Solve csc(6a+π/8) = sec(2a-π/8)</i>......(1)
Knowing that csc(x)=1/sin(x) and sec(x)=1/cos(x), we proceed to transform the equation algebraically:
1/sin(6a+π/8) = 1/cos(2a-π/8)
if sin(6a+π/8)≠0 and cos(2a-π/8)≠0, we can cross multiply to get:
cos(2a-π/8) = sin(6a+π/8) ......(2)
Using the identity
sin(θ)=cos(θ-π/2) .....(3)
we transform the right-hand-side of (2) into a cosine function.
cos(2a-π/8) = cos(6a+π/8-π/2)
cos(2a-π/8) = cos(6a-3π/8) ..... (4)
The general solution to (4) is:
±(2a-π/8) = 6a-3π/8 + 2kπ where k∈ℤ (integers)
The ± sign is applicable because cos(x)=cos(-x) for all x.
Case 1:
+(2a-π/8) = 6a-3π/8 + 2kπ
solve for a to get
a=π/16+kπ/2 ......(5)
Case 2:
-(2a-π/8) = 6a-3π/8 + 2kπ
solve for a to get
a=π/16+kπ/4 ......(6)
Since the solution (5) is a subset of (6), it would be tempting to declare the answer as given by (6).
Not so soon!
Remember we said that:
"if sin(6a+π/8)≠0 and cos(2a-π/8)≠0, we can cross multiply ...",
Now it is the time to check if those conditions are met!
By evaluating successively the solutions given by case 1 and case 2,
we will find that certain solutions of case 2 will result in:
sin(6a+π/8)=0 and cos(2a-π/8)=0. These are exactly the differences between case 2 and case 1. So we will retain case 1 as our only answer, namely:
<b>a=π/16+π/4, k∈ℤ (integers)</b>
Check my calculations and my thinking.
    
<i>Solve csc(6a+π/8) = sec(2a-π/8)</i>......(1)
Knowing that csc(x)=1/sin(x) and sec(x)=1/cos(x), we proceed to transform the equation algebraically:
1/sin(6a+π/8) = 1/cos(2a-π/8)
if sin(6a+π/8)≠0 and cos(2a-π/8)≠0, we can cross multiply to get:
cos(2a-π/8) = sin(6a+π/8) ......(2)
Using the identity
sin(θ)=cos(θ-π/2) .....(3)
we transform the right-hand-side of (2) into a cosine function.
cos(2a-π/8) = cos(6a+π/8-π/2)
cos(2a-π/8) = cos(6a-3π/8) ..... (4)
The general solution to (4) is:
±(2a-π/8) = 6a-3π/8 + 2kπ where k∈ℤ (integers)
The ± sign is applicable because cos(x)=cos(-x) for all x.
Case 1:
+(2a-π/8) = 6a-3π/8 + 2kπ
solve for a to get
a=π/16+kπ/2 ......(5)
Case 2:
-(2a-π/8) = 6a-3π/8 + 2kπ
solve for a to get
a=π/16+kπ/4 ......(6)
Since the solution (5) is a subset of (6), it would be tempting to declare the answer as given by (6).
Not so soon!
Remember we said that:
"if sin(6a+π/8)≠0 and cos(2a-π/8)≠0, we can cross multiply ...",
Now it is the time to check if those conditions are met!
By evaluating successively the solutions given by case 1 and case 2,
we will find that certain solutions of case 2 will result in:
sin(6a+π/8)=0 and cos(2a-π/8)=0. These are exactly the differences between case 2 and case 1. So we will retain case 1 as our only answer, namely:
<b>a=π/16+π/4, k∈ℤ (integers)</b>
Check my calculations and my thinking.
                    Answered by
            MathMate
            
    <b>a = π/16 + kπ/4, k∈ℤ</b> (integers) 
    
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