Expand Using Sum/Difference Formulas sin(30)

Problem

the sine of 30 raised to the composed with power

Solution

Identify the goal to represent 30 raised to the composed with power as a sum or difference of two common angles from the unit circle, such as 90 raised to the composed with power minus 60 raised to the composed with power or 45 raised to the composed with power minus 15 raised to the composed with power Using 90 raised to the composed with power minus 60 raised to the composed with power is a standard choice.
Apply the formula for the sine of a difference, which is the sine of open paren A minus B close paren equals sine A cosine B minus cosine A sine B
Substitute A equals 90 raised to the composed with power and B equals 60 raised to the composed with power into the identity.

the sine of open paren 90 raised to the composed with power minus 60 raised to the composed with power close paren equals the sine of open paren 90 raised to the composed with power close paren the cosine of 60 raised to the composed with power minus the cosine of open paren 90 raised to the composed with power close paren the sine of 60 raised to the composed with power

Evaluate the trigonometric values for the specific angles: the sine of 90 raised to the composed with power equals 1 the cosine of 60 raised to the composed with power equals one half the cosine of 90 raised to the composed with power equals 0 and the sine of 60 raised to the composed with power equals the fraction with numerator the square root of 3 and denominator 2

the sine of 30 raised to the composed with power equals 1 times one half minus 0 times open paren the fraction with numerator the square root of 3 and denominator 2 close paren

Simplify the expression to find the final numerical value.

the sine of 30 raised to the composed with power equals one half minus 0

the sine of 30 raised to the composed with power equals one half

Final Answer

the sine of 30 raised to the composed with power equals one half

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