Question
In $ \triangle$ABC, right angled at B, AB = 24 cm, BC = 7 cm. Determine:
- Sin A cos A
- Sin C cos C
✓
Answer
Let us draw a right triangle ABC.
15 cot A = 8 ...... Given
$\Rightarrow \cot A = \frac { 8 } { 15 } \Rightarrow \frac { A B } { B C } = \frac { 8 } { 15 }$
$\Rightarrow \frac { A B } { 8 } = \frac { B C } { 15 } = k ( 5 a y )$
where k is a positive number
$\Rightarrow A B = 8 k$
BC = 15k
By using the Pythagoras theorem, we have
$A C ^ { 2 } = A B ^ { 2 } + B C ^ { 2 }$
$\Rightarrow A C ^ { 2 } = ( 8 k ) ^ { 2 } + ( 15 k ) ^ { 2 } \Rightarrow A C ^ { 2 } = 64 k ^ { 2 } + 225 k ^ { 2 }$
$\Rightarrow A C = \sqrt { 289 k ^ { 2 } } \Rightarrow A C = 17 k$
Now, $\sin A = \frac { B C } { A C } = \frac { 15 k } { 17 k } = \frac { 15 } { 17 }$
and, $\sec A = \frac { A C } { A B } = \frac { 17 k } { 8 k } = \frac { 17 } { 8 }$
15 cot A = 8 ...... Given
$\Rightarrow \cot A = \frac { 8 } { 15 } \Rightarrow \frac { A B } { B C } = \frac { 8 } { 15 }$
$\Rightarrow \frac { A B } { 8 } = \frac { B C } { 15 } = k ( 5 a y )$
where k is a positive number
$\Rightarrow A B = 8 k$
BC = 15k
By using the Pythagoras theorem, we have
$A C ^ { 2 } = A B ^ { 2 } + B C ^ { 2 }$
$\Rightarrow A C ^ { 2 } = ( 8 k ) ^ { 2 } + ( 15 k ) ^ { 2 } \Rightarrow A C ^ { 2 } = 64 k ^ { 2 } + 225 k ^ { 2 }$
$\Rightarrow A C = \sqrt { 289 k ^ { 2 } } \Rightarrow A C = 17 k$
Now, $\sin A = \frac { B C } { A C } = \frac { 15 k } { 17 k } = \frac { 15 } { 17 }$
and, $\sec A = \frac { A C } { A B } = \frac { 17 k } { 8 k } = \frac { 17 } { 8 }$
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
Similar questions
Prove that the tangents drawn at the ends of a diameter of a circle are parallel.
A train travels a distance of 480 km at a uniform speed. If the speed had been $8 km / hr$ less, then it would have taken 3 hours more to cover the same distance. Find the usual speed of the train.
→A factory manufactures 120,000 pencils daily. The pencils are cylindrical in shape each of length $25\ cm$ and circumference of base as $1.5\ cm$. Determine the cost of colouring the curved surfaces of the pencils manufactured in one day at ₹ $0.05$ per $dm^2.$
→In the given figure, E is a point on the side CB produced of an isosceles triangle $A B C$ with $A B=$ AC . If $AD \perp BC$ and $EFI \perp AC$, then prove that $\triangle ABD \sim \triangle ECF$.
→Express the following as a fraction in simplest form:
$0.\overline{365}$
→$0.\overline{365}$
The angle of elevation of an aeroplane from a point on the ground is $60^{\circ}$. After a flight of $30$ seconds the angle of elevation becomes $30^{\circ}$. If the aeroplane is flying at a constant height of $3000 \sqrt{3} m$, find the speed of the aeroplane.
→Calculate the mode of the following distribution:
| Class: | 10-15 | 15-20 | 20-25 | 25-30 | 30-35 |
| Frequency: | 4 | 7 | 20 | 8 | 1 |
In the following figure, an equilateral triangle ABC of side 6cm has been inscribed a circle. Find the area of the shaded region. $\big(\text{Take }\pi=3.14\big).$
→Prove that:
$\left(\frac{1+\tan ^2 A}{1+\cot ^2 A}\right)=\frac{(1-\tan A)^2}{(1-\cot A)^2}$
→$\left(\frac{1+\tan ^2 A}{1+\cot ^2 A}\right)=\frac{(1-\tan A)^2}{(1-\cot A)^2}$
If the total surface area of a solid hemisphere is $462cm^2$, find its volume $\Big(\text{use}\ \pi=\frac{22}{7}\Big)$
→