If in a right-angled triangle, cos A = 5/13, find the value of tan A – sin A.
96/65
In a right-angled triangle, \(\cos A = \dfrac{\text{adjacent}}{\text{hypotenuse}}\). Given \(\cos A = \dfrac{5}{13}\), we take adjacent side = 5 and hypotenuse = 13.
The third (opposite) side comes from the Pythagoras theorem \(\text{opp} = \sqrt{\text{hyp}^2 - \text{adj}^2}\).
So \(\text{opp} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12\). The sides form the 5-12-13 triple.
Now write the required ratios: \(\tan A = \dfrac{\text{opp}}{\text{adj}} = \dfrac{12}{5}\) and \(\sin A = \dfrac{\text{opp}}{\text{hyp}} = \dfrac{12}{13}\).
Subtract, taking LCM of 5 and 13, which is 65: \(\tan A - \sin A = \dfrac{12}{5} - \dfrac{12}{13} = \dfrac{12 \times 13 - 12 \times 5}{65}\).
This gives \(= \dfrac{156 - 60}{65} = \dfrac{96}{65}\).
Key concept: build the full side-triangle from one ratio, then read off the others. Hence the value of \(\tan A - \sin A\) is 96/65.
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