Without using a calculator, evaluate the exact value of the trigonometric expression: \fra… — JAMB UTME

Without using a calculator, evaluate the exact value of the trigonometric expression:

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

\frac{1}{4}

Evaluating each trigonometric ratio using standard special angles (

30^\circ, 45^\circ, 60^\circ

) gives a numerator of

\left(\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} \cdot \frac{1}{2}\right) = \frac{3}{4} + \frac{1}{4} = 1

. The denominator is

\tan 45^\circ + (\tan 60^\circ)^2 = 1 + (\sqrt{3})^2 = 1 + 3 = 4

. Dividing the numerator by the denominator yields

\frac{1}{4}

Evaluate special angle values for the numerator

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

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

\cos 60^\circ = \frac{1}{2}

, and

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

Recall exact surd values for special angles

30^\circ

and

60^\circ

Simplify the numerator expression

\left(\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} \times \frac{1}{2}\right) = \frac{3}{4} + \frac{1}{4} = 1

Multiply exact surd values and add fractions.

Evaluate special angle values for the denominator

\tan 45^\circ = 1

and

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

, so

\tan^2 60^\circ = (\sqrt{3})^2 = 3

Square the exact value of

\tan 60^\circ

Simplify the denominator and divide the numerator by the denominator

\text{Denominator} = 1 + 3 = 4, \text{ so overall value} = \frac{1}{4}$.

Divide numerator result by denominator result.

Special Angle Trigonometric Ratios and Exact Value Simplification

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