Even and Odd Trig Functions Worksheet

Even and odd trigonometric functions form one of the foundations of trigonometry and functions or calculus. Some of these functions are very useful in simplifying aggregate expressions involving trigonometric components and in solving equations. Understanding whether a function is even or odd enables the students to deduce the action of the given function particularly in relation to reflection or rotation.

In this article, the reader will be given an understanding of even and odd trigonometric functions and given other relevant formulae, problems with solutions and a self-assessment test.

What are Even and Odd Trig Functions?

Even and odd functions are general concepts that apply to any function and can be applied to Discrete Mathematics, including f(x) trigonometric functions. With regards to trigonometry, these trends aid in comprehending the similarities of certain angular relationships and behaviour manifestation of functions.

• Even Functions: A function is considered even if it satisfies the condition f(-x) = -f(x) for all values of x in its domain. Even functions are symmetric about the y-axis. A common example of an even trigonometric function is the cosine function:

cos(−x)=cos(x)

• Odd Functions: A function f(x) is considered odd if it satisfies the condition f(-x) = -f(x) for all values of x in its domain. Odd functions exhibit rotational symmetry about the origin. A common example of an odd trigonometric function is the sine function:

sin(−x)= −sin(x)

Even and Odd Trig Functions

Even and Odd Trig Functions: Practice Questions with Solutions

Question 1: Determine if the function f(x) = cos(x) + cos(2x) is even, odd, or neither.

Solution:

Since both cos(x) and cos(2x) are even functions, their sum is also even. Thus, f(x) is an even function.

Question 2: Determine if the function f(x) = sin(x) - sin(3x) is even, odd, or neither.

Solution:

Since both sin(x) and sin(3x) are odd functions, their difference is also odd. Thus, f(x) is an odd function.

Question 3: Show that the function f(x) = tan(x) is odd.

Solution:

We know that (tan(−x)=−tan(x)), which satisfies the condition for odd functions.

Question 4: Determine if ( f(x) = xsin(x)) is even, odd, or neither.

Solution:

x is an odd function and sin(x) is also odd. The product of two odd functions is even, so f(x) is an even function.

Question 5: Is the function f(x) = sec(x) + csc(x) even, odd, or neither?

Solution:

sec(x) is even and sec(x) is odd. The sum of an even and odd function is neither even nor odd, so f(x) is neither.

Question 6: Simplify \cos(180^\circ - x) .

Solution:

Using the even property of cosine, \cos(180^\circ - x) = -\cos(x) .

Question 7: Simplify \sin(-45^\circ) .

Solution:

Since sine is odd, \sin(-45^\circ) = -\sin(45^\circ) = -\frac{1}{\sqrt{2}} .

Question 8: Prove that \tan(360^\circ - x) = -\tan(x) .

Solution:

\tan(-x) = -\tan(x) , and since \tan(360^\circ - x) = \tan(-x) , the equation holds.

Question 9: Evaluate \cot(-\theta) in terms of \cot(\theta) .

Solution:

Since cotangent is odd, \( \cot(-\theta) = -\cot(\theta) \).

Question 10: Is the function f(x) = \sin^2(x) + \cos^2(x) even, odd, or neither?

Solution:

Since \sin^2(x)and \cos^2(x) are both even, their sum is even.

Even and Odd Trig Functions: Worksheet

Q1. Determine if f(x) = cos(x) + sin(x) is even, odd, or neither.

Q2. Is f(x) = tan(x) - sec(x) even, odd, or neither?

Q3. Simplify \sin(90^\circ - x) .

Q4. Prove that \cos(-\theta) = \cos(\theta).

Q5. Evaluate \tan(180^\circ + x).

Q6. Determine if f(x) = x^3 \cos(x) is even, odd, or neither.

Q7. Simplify sec(−x) in terms of sec(x).

Q8. Is f(x) = x^2 \sin(x) even, odd, or neither?

Q9. Prove that \sin(-\theta) = -\sin(\theta) .

Q10. Evaluate \cot(360^\circ + x) .

Answer Key

1. Neither

2. Neither

3. cos(x)

4. True

5. tan(x)

6. Neither

7. sec(x)

8. Odd

9. True

10. cot(x)

Read More,

• Trigonometry
• Trigonometric Functions
• Trigonometry Table