Question

How do you find the 6 trigonometric functions for 90 degrees?

Answer 1

Hint: To solve these types of problems, we will use the relationship between the trigonometric ratios and some trigonometric identities. We should know the identity \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]. Also, the relationship such as, \[\tan \theta =\dfrac{\sin \theta }{\cos \theta },\csc \theta =\dfrac{1}{\sin \theta },\sec \theta =\dfrac{1}{\cos \theta }\And \cot \theta =\dfrac{1}{\tan \theta }\]. We should also know that in the first quadrant all the trigonometric ratios are positive.

Complete step by step solution:
We are given that the \[\theta ={{90}^{\circ }}\]. Here the angle is \[{{0}^{\circ }}<\theta \le {{90}^{\circ }}\]. As the angle lies in this range it means that the angle lies in the first quadrant. We know that in the first quadrant all the trigonometric ratios are positive. As \[\theta ={{90}^{\circ }}\] is a standard angle, We know the value of \[\sin {{90}^{\circ }}\] which equals 1.
We know the trigonometric identity \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\], substituting \[\theta ={{90}^{\circ }}\] in this identity, we get
\[\begin{align}
  & \Rightarrow {{1}^{2}}+{{\cos }^{2}}{{90}^{\circ }}=1 \\
 & \Rightarrow {{\cos }^{2}}{{90}^{\circ }}=0 \\
\end{align}\]
Hence, we get the value of the cosine function as \[\cos {{90}^{\circ }}=0\].
Now, we can find the other trigonometric ratios using their relationships as,
\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\], substituting the value of the ratios, we get
\[\Rightarrow \tan {{90}^{\circ }}=\dfrac{1}{0}=\infty \] or undefined
\[\Rightarrow \csc {{90}^{\circ }}=\dfrac{1}{\sin {{90}^{\circ }}}=1\]
 \[\Rightarrow \sec {{90}^{\circ }}=\dfrac{1}{\cos {{90}^{\circ }}}=\dfrac{1}{0}=\infty \]
\[\Rightarrow \cot {{90}^{\circ }}=\dfrac{0}{1}=0\]
Thus, we have found all the trigonometric ratios.

Note: To solve these types of problems, one should know the trigonometric identities and the relationship between the ratios. Here we used \[\tan \theta =\dfrac{\sin \theta }{\cos \theta },\csc \theta =\dfrac{1}{\sin \theta },\sec \theta =\dfrac{1}{\cos \theta }\And \cot \theta =\dfrac{1}{\tan \theta }\] and \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]. Also, we should know which trigonometric ratios are positive in which quadrants. In the first quadrant, all ratios are positive, in the third quadrant tangent and cotangent are positive, and in the fourth quadrant, only cosine and secant are positive.
As \[\theta ={{90}^{\circ }}\] is a standard angle, we should remember the values of its different trigonometric ratios.