Question

What are the sine, cosine and tangent of \[\theta = \dfrac{{7\pi }}{4}\] radians

Answer 1

Hint: The question is related to the trigonometry here we have to find the value of trigonometry ratios where the value of \[\theta \] , on substituting that and with the help of table of trigonometry ratios we are going to obtain the required solution for the given question.

Complete step by step solution:
The trigonometry which deals with the study of angles and the sides of the right-angled triangle. In trigonometry we have six trigonometry ratios namely, sine, cosine, tangent, cosecant, secant and cotangent.
Here we must to know about the table of trigonometry ratio, it is given as

Trigonometry ratio	\[0\]	\[\dfrac{\pi }{6}\]	\[\dfrac{\pi }{4}\]	\[\dfrac{\pi }{3}\]	\[\dfrac{\pi }{2}\]
Sine	0	\[\dfrac{1}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{{\sqrt 3 }}{2}\]	1
cosine	1	\[\dfrac{{\sqrt 3 }}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{1}{2}\]	0
tangent	0	\[\dfrac{1}{{\sqrt 3 }}\]	1	\[\sqrt 3 \]	\[\infty \]

The table is in the radians we consider this table and hence we obtain the solution.
We should know about the ASTC rule, where in the first quadrant all trigonometry ratios are positive. In the second quadrant the sine trigonometry is positive. In the third quadrant the cosine trigonometry is positive. In the fourth quadrant the tangent trigonometry is positive.
Here in this question we have \[\theta = \dfrac{{7\pi }}{4}\] radians.
Now we find the value of sine, cosine and tangent for the given radian.
The \[\sin \left( {\dfrac{{7\pi }}{4}} \right)\] , this can be written as
 \[ \Rightarrow \sin \left( {2\pi - \dfrac{\pi }{4}} \right)\]
It lies in the fourth quadrant and the sine trigonometry ratio will be negative. It is written as
 \[ \Rightarrow - \sin \left( {\dfrac{\pi }{4}} \right)\]
By the table of trigonometry ratio the value is written as
 \[ \Rightarrow - \dfrac{1}{{\sqrt 2 }}\]
The \[\cos \left( {\dfrac{{7\pi }}{4}} \right)\] , this can be written as
 \[ \Rightarrow \cos \left( {2\pi - \dfrac{\pi }{4}} \right)\]
It lies in the fourth quadrant and the cosine trigonometry ratio will be positive. It is written as
 \[ \Rightarrow \cos \left( {\dfrac{\pi }{4}} \right)\]
By the table of trigonometry ratio the value is written as
 \[ \Rightarrow \dfrac{1}{{\sqrt 2 }}\]
The \[\tan \left( {\dfrac{{7\pi }}{4}} \right)\] , this can be written as
 \[ \Rightarrow \tan \left( {2\pi - \dfrac{\pi }{4}} \right)\]
It lies in the fourth quadrant and the tangent trigonometry ratio will be negative. It is written as
 \[ \Rightarrow - \tan \left( {\dfrac{\pi }{4}} \right)\]
By the table of trigonometry ratio the value is written as
 \[ \Rightarrow - 1\]
Hence we have obtained the solution for the given question.

Note: Here we must know about the table for the trigonometry ratio. The table is the same for the degree and the radian also, the value will not change. The ASTC rule is another topic which is used here, it is abbreviated as “ALL SINE TANGENT COSINE” and in each quadrant the ratios will be positive.