Even and Odd function in Trigonometry

In this section you will learn about even and odd function in trigonometry.
Even function : A function f(x) is said to be an even function, if
f(-x) = f(x) for all x in its domain.
For example, cos(-Θ) = cos Θ
sec(-Θ) = sec Θ are even function.

Odd function : A function f(x) is said to be an odd function, if
f(-x) = -f(x) for all x in its domain.
For example, sin(-Θ) = -sin Θ
csc(-Θ) = -csc Θ
tan(-Θ) = -tan Θ
cot(-Θ)= - cot Θ are odd functions.
All the functions, including trigonometric functions, can be described as even, odd, or neither. Trigonometric even and odd functions help you in simplifying the expressions. These even and odd identities are helpful when you have an expression where the variable inside the trigonometric function is negative (like –x). The even-odd identities are as follows:
Even Function :
cos(-Θ) = cos Θ
sec(-Θ) = sec Θ
Odd Function :
sin(-Θ) = -sin Θ
csc(-Θ) = -csc Θ
tan(-Θ) = -tan Θ
cot(-Θ)= - cot Θ

Examples on even and odd function in trigonometry

Find the values of the following trigonometric ratios.
1) sin $(-480)^{0}$
Solution : We have,
sin $(-480)^{0}$
As we know that sine is an odd function.
∴ sin(-θ) = - sin θ
⇒ sin $(-480)^{0} = - sin (480)^{0}$
= - sin ( 90 x 4 + 60 )
= - sin 60
= - $\frac{\sqrt{3}}{2}$

2) cot $\frac{-15π}{4}$
Solution : We have,
cot $\frac{-15π}{4}$
As we know that cotangent is an odd function.
∴ cot(-θ) = - cot θ
⇒ cot $\frac{-15π}{4} = - cot \frac{15π}{4}$
Since $\frac{-15π}{4}= \left ( \frac{15}{4} \times 180\right )$= 675

= - cot ( 90 x 7 + 45 )
= - (-cot 45)
= - (-1)
= 1
∴ cot $\frac{-15π}{4}$ = 1

3) cos(-x)= $\frac{3}{4}$ and tan(-x) = $\frac{-\sqrt{5}}{3}$ find sin(x).

Solution : We know that ,
tan(x) = $\frac{sin(x)}{cos(x)}$
Also we know that sine is an odd function and cosine is an even function.
∴ cos(-x)= $\frac{3}{4}$ and
tangent is an odd function
∴ tan(-x) = $\frac{\sqrt{5}}{3}$
sin(x) = cos(x). tan(x)
sin(x)= $\frac{3}{4} \times \frac{\sqrt{5}}{3}$
∴ sin(x) = $\frac{\sqrt{5}}{4}$

4) If cos(x)=0.65, find cos(−x).
Solution: Since cosine is an even function,
cos(x)=cos(−x).
Therefore, cos(−x)=0.65


11th grade math

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