Question

(a) Evaluate: $\frac{s i n ^{3} 60° - t a n 30°}{c o s ^{2} 45°}$

(b) For acute angles A and B and A + 2B and 2A + B are acute if $tan (A + 2 B) = 3$ and $sin (2 A + B) = \frac{1}{2}$ , then find the measures of angles A and B.

Approach

Evaluate $\frac{s i n ^{3} 60° - t a n 30°}{c o s ^{2} 45°}$ . Substituting standard values: $sin 60° = \frac{3}{2}$ , $tan 30° = \frac{1}{3}$ , $cos 45° = \frac{1}{2}$ . Numerator: $(\frac{3}{2})^{3} - \frac{1}{3} = \frac{3 3}{8} - \frac{1}{3} = \frac{3 3 - \frac{8}{3}}{8} = \frac{9 - 8}{8 3} = \frac{1}{8 3}$ . Denominator: $(\frac{1}{2})^{2} = \frac{1}{2}$ . Result: $\frac{1/ ( 8 3 )}{1/2} = \frac{2}{8 3} = \frac{1}{4 3} = \frac{3}{12}$ .

Step-by-step working

1
Numerator: $(\frac{3}{2})^{3} - \frac{1}{3} = \frac{3 3}{8} - \frac{1}{3}$ . Converting to common denominator: $\frac{3 3 \cdot 3 - 8}{8 3} = \frac{9 - 8}{8 3} = \frac{1}{8 3}$ .
1.5 marks
2
Denominator: $cos^{2} 45° = (\frac{1}{2})^{2} = \frac{1}{2}$ . Final result: $\frac{1/ ( 8 3 )}{1/2} = \frac{1}{4 3} = \frac{3}{12}$ .
0.5 marks

Concepts used

trigonometrytrigonometric ratiosstandard anglesevaluatefind angle

Approach

Step-by-step working

Concepts used

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