Answer
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Hint: The above question is based on the concept of trigonometric functions. The main approach towards solving the above function is by knowing the secant, cosine, tangent functions can be solved by using the trigonometric identities on these functions or by using trigonometric functions whose sign depends on quadrants on the cartesian plane.
Complete step by step solution:
Trigonometric function defines the function of the angle between the both sides. It tells us the relation between the angles and sides of the right-angle triangle.
The sign of all the six trigonometric functions in the first quadrant is positive since x and y coordinates are both positive. In the second quadrant sine and cosecant are positive and in third only tangent and cotangent are positive. The fourth quadrant has only cosine function and secant function are positive.
The above given trigonometric function is secant. The opposite of secant function is cosine
function.
\[\sec \theta = \dfrac{1}{{\cos \theta }}\]
Therefore substituting the angle we get,
\[\sec 135 = \dfrac{1}{{\cos 135}}\]
The secant function has the angle 135.It can also be written as:
\[135 = 45 + 90\]
So writing it in the cosine function we get
$\cos \left( {135} \right) = \cos \left( {45 + 90} \right)$
Now further according to the quadrant rule,
\[\cos \left( {45 + 90} \right) = \sin 45 = \dfrac{{\sqrt 2 }}{2}\]
Therefore, we can write it has
\[\sec 135 = \dfrac{1}{{\cos 135}} = \dfrac{2}{{\sqrt 2 }} = \sqrt 2 \]
Therefore, we get the value of secant function.
Note: An important thing to note is that the angle 135 is written as 45+90.So the cosine function is now in the second quadrant since it is 90 +45 .The cosine function changes to sine function in the second quadrant.
Complete step by step solution:
Trigonometric function defines the function of the angle between the both sides. It tells us the relation between the angles and sides of the right-angle triangle.
The sign of all the six trigonometric functions in the first quadrant is positive since x and y coordinates are both positive. In the second quadrant sine and cosecant are positive and in third only tangent and cotangent are positive. The fourth quadrant has only cosine function and secant function are positive.
The above given trigonometric function is secant. The opposite of secant function is cosine
function.
\[\sec \theta = \dfrac{1}{{\cos \theta }}\]
Therefore substituting the angle we get,
\[\sec 135 = \dfrac{1}{{\cos 135}}\]
The secant function has the angle 135.It can also be written as:
\[135 = 45 + 90\]
So writing it in the cosine function we get
$\cos \left( {135} \right) = \cos \left( {45 + 90} \right)$
Now further according to the quadrant rule,
\[\cos \left( {45 + 90} \right) = \sin 45 = \dfrac{{\sqrt 2 }}{2}\]
Therefore, we can write it has
\[\sec 135 = \dfrac{1}{{\cos 135}} = \dfrac{2}{{\sqrt 2 }} = \sqrt 2 \]
Therefore, we get the value of secant function.
Note: An important thing to note is that the angle 135 is written as 45+90.So the cosine function is now in the second quadrant since it is 90 +45 .The cosine function changes to sine function in the second quadrant.