Question

Find m if \[\sec C = \dfrac{m}{4}\]

Answer 1

Hint: There are a total of 6 main trigonometric function sine, cosine, tangent, secant, cosecant, cotangent but if we only know the value of sine and cosine we can get the value of others easily because \[\tan A = \dfrac{{\sin A}}{{\cos A}},\sec A = \dfrac{1}{{\cos A}},\cos ecA = \dfrac{1}{{\sin A}},\cot A = \dfrac{{\cos A}}{{\sin A}}\]

Complete step by step answer:
We are given a triangle ABC whose side AB is 3 and BC is 4, now let us try to find the third side AC
By Pythagorus theorem we know that,
\[A{C^2} = A{B^2} + B{C^2}\]
So using this relation let us try to find out the value of AC
\[\begin{array}{l}
 \Rightarrow AC = \sqrt {A{B^2} + B{C^2}} \\
 \Rightarrow AC = \sqrt {{3^2} + {4^2}} \\
 \Rightarrow AC = \sqrt {9 + 16} \\
 \Rightarrow AC = \sqrt {25} \\
 \Rightarrow AC = \pm 5
\end{array}\]
Now we know that the length of a side of a triangle cannot be in negative
\[\therefore AC = 5\]
Now we know that \[\sec C = \dfrac{1}{{\cos C}}\]
So let us try to find the value of cos C
We know that \[\cos C = \dfrac{b}{h} = \dfrac{{BC}}{{AC}} = \dfrac{4}{5}\]
Now \[\sec C = \dfrac{1}{{\cos C}} = \dfrac{1}{{\dfrac{4}{5}}} = \dfrac{5}{4}\]
Which means that \[\sec C = \dfrac{m}{4} = \dfrac{5}{4}\]
And hence we get the value of m as 5.

Note: If we only knew that \[\sec C = \dfrac{h}{b}\] and then after applying pythagoras formula i could have just used the value of base and hypotenuse of the right angled triangle and still get the same answer.