What is the exact simplified value of the trigonometric expression \frac{\sin 60^\circ + \… — JAMB UTME

What is the exact simplified value of the trigonometric expression $\frac{\sin 60^\circ + \cos 30^\circ}{\tan 45^\circ + \tan 30^\circ}$ ?

\frac{3(\sqrt{3} - 1)}{2}

Substituting the exact values gives a numerator of

\sqrt{3}

and a denominator of

\frac{\sqrt{3}+1}{\sqrt{3}}

. Multiplying by the reciprocal of the denominator gives

\frac{3}{\sqrt{3}+1}

. Rationalizing the denominator by multiplying top and bottom by

(\sqrt{3}-1)

simplifies to

\frac{3(\sqrt{3}-1)}{2}

Substitute the exact standard values for each trigonometric function

\sin 60^\circ = \frac{\sqrt{3}}{2}

\cos 30^\circ = \frac{\sqrt{3}}{2}

\tan 45^\circ = 1

\tan 30^\circ = \frac{1}{\sqrt{3}}

Special angles have precise surd representations that must be used in non-calculator assessments.

Simplify the numerator and denominator separately

Numerator:

\frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = \sqrt{3}

. Denominator:

1 + \frac{1}{\sqrt{3}} = \frac{\sqrt{3} + 1}{\sqrt{3}}

Combine like terms in the numerator and find a common denominator for the terms in the denominator.

Divide the numerator by the denominator

\frac{\sqrt{3}}{\frac{\sqrt{3}+1}{\sqrt{3}}} = \frac{3}{\sqrt{3}+1}

Dividing by a fraction is equivalent to multiplying by its reciprocal.

Rationalize the denominator

\frac{3(\sqrt{3}-1)}{(\sqrt{3}+1)(\sqrt{3}-1)} = \frac{3(\sqrt{3}-1)}{3-1} = \frac{3(\sqrt{3}-1)}{2}

Multiply the numerator and denominator by the conjugate

(\sqrt{3}-1)

to remove the radical from the denominator.

Evaluation and surd simplification of special angle trigonometric expressions

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