Find the Exact Value tan(7/24)

Problem

tan((7*π)/24)

Solution

1. Identify the angle as a sum or difference of known angles. We can express (7*π)/24 as half of (7*π)/12 or as the sum of π/8 and π/6 Alternatively, use the half-angle formula for tangent where θ=(7*π)/12

2. Apply the half-angle formula tan(θ/2)=(1−cos(θ))/sin(θ) Here, θ=(7*π)/12

3. Determine the values of sin((7*π)/12) and cos((7*π)/12) using the sum formulas with (7*π)/12=π/3+π/4

4. Calculate cos(π/3+π/4)=cos(π/3)*cos(π/4)−sin(π/3)*sin(π/4)

cos((7*π)/12)=1/2⋅√(,2)/2−√(,3)/2⋅√(,2)/2

cos((7*π)/12)=(√(,2)−√(,6))/4

5. Calculate sin(π/3+π/4)=sin(π/3)*cos(π/4)+cos(π/3)*sin(π/4)

sin((7*π)/12)=√(,3)/2⋅√(,2)/2+1/2⋅√(,2)/2

sin((7*π)/12)=(√(,6)+√(,2))/4

6. Substitute these values into the half-angle formula.

tan((7*π)/24)=(1−(√(,2)−√(,6))/4)/(√(,6)+√(,2))/4

7. Simplify the fraction by multiplying the numerator and denominator by 4.

tan((7*π)/24)=(4−√(,2)+√(,6))/(√(,6)+√(,2))

8. Rationalize the denominator by multiplying by √(,6)−√(,2)

tan((7*π)/24)=((4−√(,2)+√(,6))*(√(,6)−√(,2)))/((√(,6)+√(,2))*(√(,6)−√(,2)))

9. Expand the numerator and simplify the denominator.

tan((7*π)/24)=(4√(,6)−4√(,2)−√(,12)+2+6−√(,12))/(6−2)

tan((7*π)/24)=(4√(,6)−4√(,2)−2√(,3)+8−2√(,3))/4

tan((7*π)/24)=(8−4√(,2)−4√(,3)+4√(,6))/4

10. Divide each term by 4 to reach the final exact value.

tan((7*π)/24)=2−√(,2)−√(,3)+√(,6)

Final Answer

tan((7*π)/24)=√(,6)−√(,3)−√(,2)+2

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