In $\Delta$ PQR, right-angled at $Q, P R+Q R=25 \mathrm{~cm}$ and $P Q=5 \mathrm{~cm}$. Determine the values of $\sin P, \cos P$ and $\tan \bar{P}$.

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Precalculus

NCERT Class 10 - Math

Chapter 8

Introduction to Trigonometry

Section 1

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So in the given question we have we have a right angled triangle peak. You are which is having the right angle at Q. So it is right angle that you. And we are told that in the triangle when we add the lengths of the sides tear and Cuba we get 25 and PQ as having a length of five centimeters. So this is what we are given. And we are now told to find the values of sign P. Cosby and 10 of P. So when we have a triangle, a right angled triangle. And we need to find the Technimetrics ratios related to a specific angle theater. Then what we can do is we can take this as the opposite side to the angle theater. This side is the largest inside to angle theater. And this of course is the hyper news of this by triangle. Right? So we can take the ratios scientific to us, scientists to us the opposite side do I don't bite the high partners. We can take the cause of the angle theta as the length of the are just inside divided by high partners. And we can take the time off the town of data. Turn of the to as the opposite side. The opposite side. Let's write it all here. So you can take and of data as the opposite side divided by the hardest inside. So this is what we have. Right? So we are going to use these relations are these ratios in order to find the values of ST B. Cosby and dan P. Over here. So first of all, what we need to do is we are told that Pr press cuba Is equal to 25. Right? So on taking on squaring both sides of the situation, we would have pr plus Cuba squared is equal to 25 squared, which is equal to 625. And when we expand the left hand side we have P. R squared plus cuba squared. Place two. Hear names cuba As equal do 6 25. Right? And from this we are also given that he through the spine, right PQ. S equal to fine next. What we can do is we can take the equation that according to Peta barest era according to by categories Tierra in a triangle. The hyper news, the length of the hypotenuse squared can be found to be equal to the sum of squares. Off squares are the The other two sides. The other two sides of the triangle. Yeah. So what we can take from this relation is that we can say that we Q squared plus cuba squared is equal to P. R squared. Right? So now what we can write us since pre-2 squared is equal to PQ is equal to five. We can write Pierre squared minus you were squared is equal to The square of PQ which is 25. And from this weekend, right? And this weekend right that P R is equal to P R squared is equal to 25 plus cuba squared 25 plus cuba squared. Which we can substitute in this question. Right? It's called the situation equation # one. So what we can write as we have P R squared minus your square is equal to 25. So we can either substitute it in that equation or we can just. Right. He asked where minus keywords where as Pr minus cuba times beyond plus cuba. Since we can write a squared minus B squared in the form, A plus B times A minus B. We can use this form. And the question, we already have that pr plus cuba is equal to 25 which means we can right, we can write pr minus cuba Times 25 is equal to 25, Which means pr minus Cuba is equal to 25, divided by 25 Which is equal to one. So now what we have is to relations right? The relation that is given in the question which is peer plus Cuba is equal to 25. And the relation that we just got which is peered minus Cuba is equal to want. So when we add this equation, what we would get us to P. R. Is equal to 26 Or from this weekend right? PR. is equal to 13. So if PR is equal to 13 we can write cuba as equal to 25 -13, which is equal to 12. So now we have got all the sides of the triangle, right? We have Pr which is 13 cuba being P. Q R P R 13 P Q. S five and cuba is 12. And now we need to find the trigonometry ratios related to the and repeat. So we don't have to do we don't have to expand this like this. Right? So we don't have to expand this like this. We can just directly take the by the greatest era and substitute the given values in the relation that we get from pythagoras theorem in order to find decides here. And you're So once we have found all the sides now we can use the economic ratios that assigned P. Is equal to the opposite side, to the angle P divided by the hyper news. Cosby is equal to the ar just inside to angle P, divided by the hypothalamus and turn P. Is equal to the opposite side, divided by they are just inside. So these are what we need to find over here, right sign P cost p. and 10 p. So these are the values of the economic trick ratios that we needed to find. I hope you understood the method. Thank you.

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