Question

If triangle ABC is a right angle triangle at C then the value of sec(A + B) is
$\begin{align}
  & \text{a) 0} \\
 & \text{b) 1} \\
 & \text{c) }\dfrac{2}{\sqrt{3}} \\
 & \text{d) Not defined} \\
\end{align}$

Answer 1

Hint: Now we know that the sum of interior angles of triangles is always equal to ${{180}^{\circ }}$ . Since we have measure of angle C, we can use this property to find the measure of angle A + Angle B. Once we have angle (A + B) we can easily find its sec value.

Complete step by step answer:
Now consider a right angle triangle ABC such that angle C is a right angle which means measure of angle C is 90 degrees

Now in triangle ABC we have $\angle ACB={{90}^{\circ }}$
Now we know that the sum of the interior angles in triangles is always equal to ${{180}^{\circ }}$ .
Hence using the above property in our given triangle ABC we get
$\angle ABC+\angle ACB+\angle CBA={{180}^{\circ }}$
Now substituting the value of angle ACB which is 90 in the above equation we get
$\angle ABC+{{90}^{\circ }}+\angle CBA={{180}^{\circ }}$
Now subtracting 90 degrees on both sides we get
$\begin{align}
  & \angle ABC+\angle CBA={{180}^{\circ }}-{{90}^{\circ }} \\
 & \Rightarrow \angle ABC+\angle CBA={{90}^{\circ }} \\
\end{align}$
Now we can write angle ABC as angle B and angle CBA as angle B
$\therefore \angle A+\angle B={{90}^{\circ }}...................\left( 1 \right)$
Now we want to find out sec(A + B).
Now from equation (1) we get sec(A + B) = sec(90)
Now from the trigonometric table we know that the value $\sec {{90}^{\circ }}$ is not defined.
Hence we have sec(A + B) is not defined.

So, the correct answer is “Option D”.

Note: Note that the trigonometric ratios sec is nothing but inverse of cos. Which means $\sec \theta =\dfrac{1}{\cos \theta }$ . Now we know that $\cos {{90}^{\circ }}=0$ hence we will get $\sec {{90}^{\circ }}=\dfrac{1}{0}$ but we know that any number divided by 0 is not defined.