Question

If triangle ABC is right angled at C, then the value of \[\sec \left( {A + B} \right)\] is
(A) 0
(B) 1
(C) $\dfrac{2}{{\sqrt 3 }}$
(D) not defined

Answer 1

Hint: This problem comes under application of trigonometric Identities on triangle bases sums. Here we asked to find the angle of A and B and given that right angled at C. By using angle sum property we used to find the angle of A and B and then with trigonometric angle and ratio identities we obtain \[\sec \left( {A + B} \right)\] and by complete step by step explanation.

Formula used:
Angle sum property

In a triangle sum of three sides of angle is equal to \[{180^ \circ }\]
That is, \[\sec {90^ \circ } = \dfrac{1}{0}\]
\[\sec \theta = \dfrac{1}{{\cos \theta }}\]
\[\cos {90^ \circ } = 0\]

Complete step-by-step answer:
Given that in $\vartriangle ABC$, right angled at C,
Therefore $\angle C = {90^ \circ }$
By using angle sum property of triangle mentioned in formula used and substitute $\angle C = {90^ \circ }$, we get
$\Rightarrow$\[\angle A + \angle B + {90^ \circ } = {180^ \circ }\]
By separating numerals, we get
$\Rightarrow$\[\angle A + \angle B = {180^ \circ } - {90^ \circ }\]
Subtracting we get,
$\Rightarrow$\[\angle A + \angle B = {90^ \circ }\]
Now, Substitute \[\angle A + \angle B = {90^ \circ }\]in \[\sec \left( {A + B} \right)\], we get
$\Rightarrow$\[\sec (A + B) = \sec {90^ \circ }\]
Now using trigonometric ratio formula mentioned in formula used we get,
$\Rightarrow$\[\sec {90^ \circ } = \dfrac{1}{{\cos {{90}^ \circ }}}\]
Again using trigonometric angle value mentioned in formula used, we get
$\Rightarrow$\[\sec {90^ \circ } = \dfrac{1}{0}\] which is not defined
Therefore, \[\sec \left( {A + B} \right) = \] not defined which means infinity

Option D is the correct answer.

Note: This kind problem needs to know well about triangle and trigonometric identities, ratios and angles. The sum mainly depends on the angle sum property of the triangle in the process of making a solution and then it has to be substituted in trigonometric angle and then with basic mathematical calculation we find the answer. This is a multiple choice based question so we need to find the correct option. The sum of angles in a triangle is equal to 180.