Find the Exact Value arcsin(sin((5pi)/7))

Problem

arcsin(sin((5*π)/7))

Solution

1. Identify the range of the principal inverse sine function, which is [−π/2,π/2]

2. Determine if the input angle (5*π)/7 lies within this range. Since (5*π)/7>π/2 (because 5/7>3.5/7, the angle is in the second quadrant and outside the principal range.

3. Apply the trigonometric identity sin(θ)=sin(π−θ) to find a reference angle within the principal range that has the same sine value.

4. Calculate the equivalent angle:

π−(5*π)/7=(7*π)/7−(5*π)/7

(7*π−5*π)/7=(2*π)/7

5. Verify that (2*π)/7 is within the interval [−π/2,π/2] Since 0<2/7<1/2 the condition is satisfied.

6. Evaluate the expression using the property arcsin(sin(x))=x for x∈[−π/2,π/2]

Final Answer

arcsin(sin((5*π)/7))=(2*π)/7

Want more problems? Check here!