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Question

$alpha$ is a root of $25 cos to the power of 2 end exponent invisible function application theta plus 5 cos invisible function application theta minus 12 equals 0 comma fraction numerator pi over denominator 2 end fraction less than alpha less than pi comma$ then $blank sin invisible function application 2 alpha blank$ is equal to

$\frac{24}{25}$
$- \frac{24}{25}$
$\frac{13}{18}$
$- \frac{13}{18}$

The correct answer is: $negative fraction numerator 24 over denominator 25 end fraction$

Since $alpha$ is a root of $25 cos to the power of 2 end exponent invisible function application theta plus 5 cos invisible function application theta minus 12 equals 0$
$therefore 25 cos to the power of 2 end exponent invisible function application alpha plus 5 cos invisible function application alpha minus 12 equals 0 blank rightwards double arrow cos invisible function application alpha equals fraction numerator negative 5 plus-or-minus square root of 25 plus 1200 end root over denominator 50 end fraction equals negative fraction numerator 4 over denominator 5 end fraction open square brackets because fraction numerator pi over denominator 2 end fraction less than alpha less than pi close square brackets$
$a n d sin invisible function application alpha equals square root of 1 minus fraction numerator 16 over denominator 25 end fraction end root equals fraction numerator 3 over denominator 5 end fraction semicolon t h e r e f o r e comma sin invisible function application 2 alpha equals 2 sin invisible function application alpha cos invisible function application alpha equals 2 open parentheses fraction numerator 3 over denominator 5 end fraction close parentheses open parentheses fraction numerator negative 4 over denominator 5 end fraction close parentheses equals fraction numerator negative 24 over denominator 25 end fraction$

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