Answer

Verified

360.9k+ views

**Hint:**We will first draw the figure from the given details in the question and then we will apply the Pythagoras theorem to find the value of the missing side. And after this we will see the point from which we need to find the values and according to that we will select our perpendicular and hypotenuse.

**Complete step by step answer:**

Before proceeding with the question we should know the concept of Pythagoras theorem and right angled triangle.

A Right-angled triangle is one of the most important shapes in geometry and is the basics of trigonometry. A right-angled triangle is the one which has 3 sides, “base” “hypotenuse” and “height” with the angle between base and height being 90 degrees.

The Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“. The sides of this triangle have been named as Perpendicular, Base and Hypotenuse. Here, the hypotenuse is the longest side, as it is opposite to the angle 90 degree.

So we will first draw the figure from the given details in the question.

Now we will use Pythagoras theorem and from the figure we get,

\[\Rightarrow P{{R}^{2}}=P{{Q}^{2}}+Q{{R}^{2}}........(1)\]

Now substituting the known values in equation (1) we get,

\[\Rightarrow P{{R}^{2}}={{4}^{2}}+{{3}^{2}}........(2)\]

Now squaring the terms in the right hand side of the equation (2) and then adding we get,

\[\Rightarrow P{{R}^{2}}=16+9=25........(3)\]

Now taking the square root on both sides in equation (3) we get,

\[\Rightarrow PR=5........(4)\]

From the figure we can see that sin C is PQ (perpendicular) divided by PR (hypotenuse). Hence using this information we get,

\[\Rightarrow \sin P=\dfrac{perpendicular}{hypotenuse}=\dfrac{QR}{PR}=\dfrac{3}{5}\]

Now, similarly, we can calculate the value of cos Q as follows

\[\Rightarrow \cos Q=\dfrac{base}{hypotenuse}=\dfrac{PQ}{PR}=\dfrac{4}{5}\]

Now we know that \[\text{sec}\,P=\dfrac{1}{\cos P}\]. Hence substituting the value of cos P in this we get,

\[\Rightarrow \text{sec}\,P=\dfrac{5}{4}\]

Hence the value of \[\text{sec}\,P\] is \[\dfrac{5}{4}\].

Also, we can calculate the value of cos R as follows

\[\Rightarrow \cos R=\dfrac{base}{hypotenuse}=\dfrac{QR}{PR}=\dfrac{3}{5}\]

Now we know that \[\text{sec}\,R=\dfrac{1}{\cos R}\]. Hence substituting the value of cos R in this we get,

\[\Rightarrow \text{sec}\,R=\dfrac{5}{3}\]

**Hence the value of \[\text{sec}\,C\] is \[\dfrac{5}{3}\].**

**Note:**Always remember that if we have right angle triangle, then using Pythagoras theorem, we can find any of the side of triangle as it states that \[{{H}^{2}}={{P}^{2}}+{{B}^{2}}\], where H is hypotenuse, P is perpendicular and B is base of triangle and also $\operatorname{sinA}=\dfrac{Perpendicular}{Hypotenuse}$ , \[\operatorname{cosA}=\dfrac{Base}{Hypotenuse}\] and \[\tan A=\dfrac{Perpendicular}{Base}\]. While simplifying numerical terms do calculation carefully as it will change the final answer and will make the solution more complex.

Recently Updated Pages

The branch of science which deals with nature and natural class 10 physics CBSE

Three beakers labelled as A B and C each containing 25 mL of water were taken A small amount of NaOH anhydrous CuSO4 and NaCl were added to the beakers A B and C respectively It was observed that there was an increase in the temperature of the solutions contained in beakers A and B whereas in case of beaker C the temperature of the solution falls Which one of the following statements isarecorrect i In beakers A and B exothermic process has occurred ii In beakers A and B endothermic process has occurred iii In beaker C exothermic process has occurred iv In beaker C endothermic process has occurred

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Trending doubts

How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Distinguish between the reserved forests and protected class 10 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Give simple chemical tests to distinguish between the class 12 chemistry CBSE

Difference Between Plant Cell and Animal Cell

Which of the following books is not written by Harshavardhana class 6 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

In which states of India are mango showers common What class 9 social science CBSE

What Made Mr Keesing Allow Anne to Talk in Class class 10 english CBSE