Question

What are the cofunction identities and reflection properties for trigonometric functions?

Answer 1

Hint: In this question, we are asked about the cofunction identities and reflection properties of trigonometric functions. So, initially we look at the definitions of cofunction identities and reflection properties. Then, we write the identities and properties one by one.

Complete step by step answer:
Let us start solving the question.
In this question, we are asked about the cofunction identities and reflection properties of trigonometric functions. So, initially we look at the definitions of cofunction identities and reflection properties. Cofunction identities are the identities which show the relationship among the trigonometric functions like sine, cosine, tangent, cotangent, cosecant and secant. In trigonometry, complementary angles are defined as the angles having sum \[{{90}^{\circ }}\].
Reflection properties are defined as the change in sign on value of trigonometric function if angle is reflected to the other side of \[{{0}^{\circ }}\].
Cofunction identities:
\[\sin \left( \dfrac{\pi }{2}-\theta \right)=\cos \theta \]
\[\cos \left( \dfrac{\pi }{2}-\theta \right)=\sin \theta \]
\[\tan \left( \dfrac{\pi }{2}-\theta \right)=\cot \theta \]
\[\cot \left( \dfrac{\pi }{2}-\theta \right)=\tan \theta \]
\[\cos ec\left( \dfrac{\pi }{2}-\theta \right)=\sec \theta \]
\[\sec \left( \dfrac{\pi }{2}-\theta \right)=\operatorname{cosec}\theta \]
As we can see in the above relations that cofunction identities is basically the complementary relationship between trigonometric functions.
Reflection properties:
 \[\sin \left( -\theta \right)=-\sin \theta \]
\[\cos \left( -\theta \right)=+\cos \theta \]
\[\tan \left( -\theta \right)=-\tan \theta \]
\[\cot \left( -\theta \right)=-\cot \theta \]
\[\cos ec\left( -\theta \right)=-\cos ec\theta \]
\[sec\left( -\theta \right)=+sec\theta \]
The above equations are called reflection properties when the reflection is taken from the x-axis between the first and fourth quadrant of the graph.
These both cofunction identities and reflection properties are very useful in solving complex trigonometric problems.

Note: If we consider the value of trigonometric functions quadrant wise then reflection properties are considered in a way that what will be the value of trigonometric function if we move towards the fourth quadrant from x-axis. Conventionally, moving towards the first quadrant is considered as the positive side and moving towards the fourth quadrant is considered as the negative side. There are some other properties also like reciprocal identities and inverse functions which become very useful in solving problems.