Question

If $tan A = 3$ ; where A is an acute angle, then find the value of $\frac{s i n ^{2} A}{1 + c o s ^{2} A}$ .

Approach

Given $tan A = 3$ , so $A = 60°$ . Substitute into $\frac{s i n ^{2} A}{1 + c o s ^{2} A} = \frac{s i n ^{2} 60°}{1 + c o s ^{2} 60°} = \frac{( \frac{3}{2} ) ^{2}}{1 + ( \frac{1}{2} ) ^{2}} = \frac{\frac{3}{4}}{\frac{5}{4}} = \frac{3}{5}$ .

Step-by-step working

1
$tan A = 3 = tan 60° \Rightarrow A = 60°$ .
0.5 marks
2
Substitute $A = 60°$ : $\frac{s i n ^{2} 60°}{1 + c o s ^{2} 60°} = \frac{( \frac{3}{2} ) ^{2}}{1 + ( \frac{1}{2} ) ^{2}}$ .
1 mark
3
Simplify: $\frac{\frac{3}{4}}{1 + \frac{1}{4}} = \frac{\frac{3}{4}}{\frac{5}{4}} = \frac{3}{5}$ .
0.5 marks

Concepts used

trigonometric ratiostansincosfind value

Approach

Step-by-step working

Concepts used

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