Simplify sin(pi/9)+cos(pi/9)

Problem

sin(π/9)+cos(π/9)

Solution

1. Apply the harmonic addition identity to rewrite the expression in the form R*sin(x+α) For an expression a*sin(x)+b*cos(x) the identity is √(,a2+b2)*sin(x+α) where tan(α)=b/a

2. Identify the coefficients a and b from the expression. Here, a=1 and b=1

3. Calculate the amplitude R

R=√(,1+1)

R=√(,2)

4. Determine the phase shift α Since a=1 and b=1 we find α such that tan(α)=1/1

α=π/4

5. Substitute the values into the identity formula.

sin(π/9)+cos(π/9)=√(,2)*sin(π/9+π/4)

6. Find a common denominator to add the fractions inside the sine function. The common denominator for 9 and 4 is 36

π/9+π/4=(4*π)/36+(9*π)/36

(4*π)/36+(9*π)/36=(13*π)/36

7. Write the simplified expression using the combined angle.

√(,2)*sin((13*π)/36)

Final Answer

sin(π/9)+cos(π/9)=√(,2)*sin((13*π)/36)

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