In triangle PQR , right - angled at Q , PR + QR =25, cm and PQ =5, cm . Determine values of sin P, c

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8. Introduction to Trigonometry

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In $\triangle PQR ,$ right $-$ angled at $Q , PR + QR =25\, cm$ and $PQ =5\, cm .$ Determine the values of $\sin P, \cos P$ and $\tan P$.

Option A

Option B

Option C

Option D

Solution

Given that, $PR + QR =25$

$PQ =5$

Let $PR$ be $x$.

Therefore, $QR =25-x$

Applying Pythagoras theorem in $\triangle PQR$, we obtain

$PR ^{2}= PQ ^{2}+ QR ^{2}$

$x^{2}=(5)^{2}+(25-x)^{2}$

$x^{2}=25+625+x^{2}-50 x$

$50 x=650$

$x=13$

Therefore, $PR =13 \,cm$

$Q R=(25-13) \,cm =12\, cm$

$\sin P =\frac{\text { Side opposite to } \angle P }{\text { Hypotenuse }}=\frac{ QR }{ PR }=\frac{12}{13}$

$\cos P =\frac{\text { Side adjacent to } \angle P }{\text { Hypotenuse }}=\frac{ PQ }{ PR }=\frac{5}{13}$

$\tan P =\frac{\text { Side opposite to } \angle P }{\text { Side adjacent to } \angle P }=\frac{ QR }{ PQ }=\frac{12}{5}$

Standard 10

Mathematics

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hard

In $\triangle PQR ,$ right $-$ angled at $Q , PR + QR =25\, cm$ and $PQ =5\, cm .$ Determine the values of $\sin P, \cos P$ and $\tan P$.

Solution

Similar Questions

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Prove that $\frac{\cot A-\cos A}{\cot A+\cos A}=\frac{\operatorname{cosec} A-1}{\operatorname{cosec} A+1}$

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Evaluate the following:

$\frac{\sin 30^{\circ}+\tan 45^{\circ}-\operatorname{cosec} 60^{\circ}}{\sec 30^{\circ}+\cos 60^{\circ}+\cot 45^{\circ}}$

In triangle $ABC ,$ right -angled at $B ,$ if $\tan A =\frac{1}{\sqrt{3}},$ find the value of:

$(i)$ $\sin A \cos C+\cos A \sin C$

$(ii)$ $\cos A \cos C-\sin A \sin C$

Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

$\left(\frac{1+\tan ^{2} A}{1+\cot ^{2} A}\right)=\left(\frac{1-\tan A}{1-\cot A}\right)^{2}=\tan ^{2} A$