Question
Given: $4\cot A = 3$;find :$(i)\sin A;(ii)\sec A;(iii)\operatorname{cosec}^2A - \cot^2A.$
✓
Answer
Consider the diagram below :
$4 \cot A =3$
$\cot A =\frac{3}{4}$
i.e. $\frac{\text { base }}{\text { perpendicular }}=\frac{3}{4}$
$\Rightarrow \frac{ AB }{ BC }=\frac{3}{4}$
Therefore if length of $AB = 3x,$ length of $BC = 4x$
Since
$AB^2 + BC^2 = AC^2 \dots...[$ Using Pythagoras Theorem $]$
$(3x)^2 + (4x)^2 = AC^2$
$AC^2 = 9x^2 + 16x^2 = 25x^2$
$\therefore AC = 5x \dots...($ hypotenuse $)$
$(i)\sin A =\frac{\text { perpendicular }}{\text { hypotenuse }}=\frac{4 x}{5 x}=\frac{4}{5}$
$(ii)\sec A =\frac{\text { hypotenuse }}{\text { base }}=\frac{ AC }{ AB }=\frac{5 x}{3 x}=\frac{5}{3}$
$(iii) \operatorname{cosec} A =\frac{\text { hypotenuse }}{\text { perpendicular }}=\frac{ AC }{ BC }=\frac{5 x}{4 x}=\frac{5}{4}$
$\cot A =\frac{3}{4}$
$\operatorname{cosec}^2A – \cot^2 A$
$=\left(\frac{5}{4}\right)^2-\left(\frac{3}{4}\right)^2$
$=\frac{25-9}{16}$
$=\frac{16}{16}$
$= 1$
$4 \cot A =3$
$\cot A =\frac{3}{4}$
i.e. $\frac{\text { base }}{\text { perpendicular }}=\frac{3}{4}$
$\Rightarrow \frac{ AB }{ BC }=\frac{3}{4}$
Therefore if length of $AB = 3x,$ length of $BC = 4x$
Since
$AB^2 + BC^2 = AC^2 \dots...[$ Using Pythagoras Theorem $]$
$(3x)^2 + (4x)^2 = AC^2$
$AC^2 = 9x^2 + 16x^2 = 25x^2$
$\therefore AC = 5x \dots...($ hypotenuse $)$
$(i)\sin A =\frac{\text { perpendicular }}{\text { hypotenuse }}=\frac{4 x}{5 x}=\frac{4}{5}$
$(ii)\sec A =\frac{\text { hypotenuse }}{\text { base }}=\frac{ AC }{ AB }=\frac{5 x}{3 x}=\frac{5}{3}$
$(iii) \operatorname{cosec} A =\frac{\text { hypotenuse }}{\text { perpendicular }}=\frac{ AC }{ BC }=\frac{5 x}{4 x}=\frac{5}{4}$
$\cot A =\frac{3}{4}$
$\operatorname{cosec}^2A – \cot^2 A$
$=\left(\frac{5}{4}\right)^2-\left(\frac{3}{4}\right)^2$
$=\frac{25-9}{16}$
$=\frac{16}{16}$
$= 1$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
More "[4 marks sum]" from Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals]→Trigonometrical Ratios [Sine, Consine, Tangent of an Angle and their Reciprocals] — all question types→MATHEMATICS STD 9 question papers→ICSE Board STD 9 question papers→ICSE Board question paper generator→
Similar questions
From Delhi station, if we buy $2$ tickets for station $A$ and $3$ tickets for station $B,$ the total cost is $Rs. 77$. But if we buy $3$ tickets for station $A$ and $5$ tickets for station $B$, the total cost is $Rs. 124$. What are the fares from Delhi to station $A$ and to station $B$ ?
→A man leaves half his property to his wife, one$-$third to his son and the remaining to his daughter. If the daughter's share is $Rs. 15000$, how much money did the man leave? How much money did his wife get?
→Draw parallelogram $\text{ABCD}$ with the following data:$A B=6 \ cm , A D=5 \ cm$ and $\angle D A B=45^{\circ}$.Let $A C$ and $D B$ meet in $O$ and let $E$ be the mid$-$point of $B C$. Join $O E$.Prove that:$(i) OE \| AB;(ii) OE =\frac{1}{2} AB$.
→On a certain sum of money, invested at the rate of $10$ percent per annum compounded annually, the interest for the first year plus the interest for the third year is $Rs. 2,652$. Find the $\sum.$
→Construct a cumulative frequency distribution table from the frequency table given below:
→| Class Interval | Frequency |
| $1 - 10$ | $12$ |
| $11 - 20$ | $18$ |
| $21 - 30$ | $23$ |
| $31 - 40$ | $15$ |
| $41 - 50$ | $10$ |
$\text{PQR}$ is an isosceles triangle with $PQ = PR = 10 \ cm$ and $QR = 12 \ cm$. Find the length of the perpendicular from $P$ to $QR$.
→Solve for $x : \log (x + 5) + \log (x - 5) = 4 \log 2 + 2 \log 3$
→If $AP$ bisects $\angle BAC$ and $M$ is any point on $AP,$ prove that the perpendiculars drawn from $M$ to $AB$ and $AC$ are equal.
→In the given figure, $\angle ABC =66^{\circ}, \angle DAC =38^{\circ}$. CE is perpendicular to AB and AD is perpendicular to BC .
Prove that: $CP > AP$.
→Prove that: $CP > AP$.
The perimeter of a semi-circular metallic plate is 86.4 cm. Calculate the radius and area of the plate.