What is the exact numerical value of the trigonometric expression \frac{6\sin^2 60^\circ -… — JAMB UTME

What is the exact numerical value of the trigonometric expression $\frac{6\sin^2 60^\circ - 2\cos^2 45^\circ}{\tan^2 30^\circ + \sec^2 45^\circ}$ ?

Cevap: 1.5

1.5

Substituting the exact special angle values gives a numerator of

6\left(\frac{3}{4}\right) - 2\left(\frac{1}{2}\right) = \frac{7}{2}

and a denominator of

\frac{1}{3} + 2 = \frac{7}{3}

. Dividing

\frac{7}{2}

\frac{7}{3}

yields

\frac{3}{2} = 1.5

Substitute the exact values for the trigonometric ratios of the special angles.

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

\cos 45^\circ = \frac{1}{\sqrt{2}}

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

, and

\sec 45^\circ = \sqrt{2}

Exact surd forms for special angles

30^\circ

45^\circ

, and

60^\circ

must be used.

Evaluate and simplify the numerator expression

6\sin^2 60^\circ - 2\cos^2 45^\circ

6\left(\frac{3}{4}\right) - 2\left(\frac{1}{2}\right) = \frac{9}{2} - 1 = \frac{7}{2}

Square each trigonometric ratio first, multiply by the coefficients, and then subtract.

Evaluate and simplify the denominator expression

\tan^2 30^\circ + \sec^2 45^\circ

\left(\frac{1}{\sqrt{3}}\right)^2 + (\sqrt{2})^2 = \frac{1}{3} + 2 = \frac{7}{3}

Square each trigonometric ratio and simplify the sum into a single improper fraction.

Divide the numerator result by the denominator result.

\frac{7/2}{7/3} = \frac{7}{2} \times \frac{3}{7} = \frac{3}{2} = 1.5

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

Evaluation of Trigonometric Expressions using Special Angles

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