What is the exact numerical value of the trigonometric expression \frac{12 \sin 30^\circ \… — JAMB UTME

What is the exact numerical value of the trigonometric expression

\frac{12 \sin 30^\circ \cos 30^\circ \tan 60^\circ}{\sin^2 45^\circ + \cos^2 45^\circ + \tan^2 60^\circ}

Answer: 2.25

The exact numerical value of the expression is 2.25.

Substituting the exact values

\sin 30^\circ = \frac{1}{2}

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

, and

\tan 60^\circ = \sqrt{3}

into the numerator gives

12 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} \times \sqrt{3} = 9

. Using the Pythagorean identity

\sin^2 45^\circ + \cos^2 45^\circ = 1

and

\tan^2 60^\circ = 3

, the denominator evaluates to

1 + 3 = 4

. Dividing 9 by 4 yields the exact value of 2.25.

Evaluate the trigonometric ratios for special angles and apply trigonometric identities.

\sin 30^\circ = \frac{1}{2}

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

\tan 60^\circ = \sqrt{3}

, and

\sin^2 45^\circ + \cos^2 45^\circ = 1

Standard special angle values and the fundamental Pythagorean identity

\sin^2 x + \cos^2 x = 1

simplify the expression.

Substitute these values into the numerator of the expression.

Numerator

= 12 \times \frac{1}{2} \times \frac{\sqrt{3}}{2} \times \sqrt{3} = 9

Multiplying the terms:

\sqrt{3} \times \sqrt{3} = 3

, and

12 \times \frac{1}{4} \times 3 = 9

Substitute these values into the denominator of the expression.

Denominator

= 1 + (\sqrt{3})^2 = 1 + 3 = 4

The sum

\sin^2 45^\circ + \cos^2 45^\circ = 1

combined with

\tan^2 60^\circ = 3

equals 4.

Divide the calculated numerator by the denominator.

\frac{9}{4} = 2.25

Dividing 9 by 4 gives the final decimal value 2.25.

Evaluation of trigonometric expressions using special angles (30°, 45°, 60°) and fundamental identities

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