Find the Exact Value sin(5)

Problem

sin(5^∘)

Solution

1. Identify the expression as the sine of 5 degrees. Since 5^∘ is not a standard angle, we use the half-angle formula or angle subtraction formulas.

2. Apply the half-angle formula for sine, which is sin(θ/2)=√(,(1−cos(θ))/2) for an angle in the first quadrant.

3. Substitute θ=10^∘ into the formula to get sin(5^∘)=√(,(1−cos(10^∘))/2)

4. Use the identity for cos(10^∘) derived from the triple-angle formula or other radical expressions. A common exact form for sin(5^∘) involves nested radicals derived from sin(18^∘−15^∘) or sin(15^∘−10^∘)

5. Express the value using the combination of 15^∘ and 10^∘ or 18^∘ and 13^∘ is complex; however, the standard exact radical form is derived from sin(15^∘−10^∘) or similar.

6. Simplify the expression to its standard radical form.

Final Answer

sin(5^∘)=√(,2)/4*(√(,3)*sin(10^∘)−cos(10^∘))

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