Find the Exact Value sec(pi/10)

Problem

sec(π/10)

Solution

1. Identify the relationship between the secant function and the cosine function using the reciprocal identity sec(θ)=1/cos(θ)

sec(π/10)=1/cos(π/10)

2. Determine the exact value of cos(π/10) which corresponds to cos(18) This value is derived from the geometry of a regular pentagon or the golden ratio.

cos(π/10)=√(,10+2√(,5))/4

3. Substitute the cosine value into the reciprocal expression.

sec(π/10)=1/√(,10+2√(,5))/4

4. Simplify the fraction by multiplying by the reciprocal of the denominator.

sec(π/10)=4/√(,10+2√(,5))

5. Rationalize the denominator by multiplying the numerator and denominator by √(,10−2√(,5))

sec(π/10)=(4√(,10−2√(,5)))/√(,(10+2√(,5))*(10−2√(,5)))

6. Simplify the expression inside the square root in the denominator using the difference of squares.

sec(π/10)=(4√(,10−2√(,5)))/√(,100−20)

7. Evaluate the square root in the denominator.

sec(π/10)=(4√(,10−2√(,5)))/√(,80)

8. Simplify the radical √(,80) as 4√(,5)

sec(π/10)=(4√(,10−2√(,5)))/(4√(,5))

9. Cancel the common factor of 4 and simplify the remaining radical expression.

sec(π/10)=√(,(10−2√(,5))/5)

10. Distribute the division by 5 inside the square root.

sec(π/10)=√(,2−(2√(,5))/5)

Final Answer

sec(π/10)=√(,2−(2√(,5))/5)

Want more problems? Check here!