Even-Odd Identities

Even-odd identities are mathematical relationships that describe how the sine and cosine functions behave based on angle.

• Even Functions: A function f(x) is called even if f(−x) = f(x). This means that the function is symmetrical about the y-axis. The cosine function is an example of an even function.
• Odd Functions: A function f(x) is called odd if f(−x) = −f(x). This means that the function is symmetrical about the origin. The sine function is an example of an odd function.

In trigonometry, there are many types of identities, including Pythagorean identities, even-odd identities, reciprocal identities, and sum and difference identities.

Even-Odd Identities in Trigonometry

In trigonometry, some functions behave as even functions, while others behave as odd. Here’s how they break down:

Proof of Even-Odd Identities

As we know, -x (if x < 90°) lies in the fourth quadrant, and only positive values in that quadrant are cos and sec.

• cos⁡(−x) = cos⁡(x) and sec⁡(−x) = sec⁡(x)

Read More - Unit Circle

Solved Questions on Even-Odd Identities

Question 1: Evaluate cos⁡(−60°) + sin⁡(−30°).

Solution:

First, simplify each term using the appropriate identities:

• For cos⁡(−60°): cos⁡(−60°) = cos⁡(60°) = 1/2
• For sin⁡(−30°): sin⁡(−30°) = −sin⁡(30°) = −1/2​

Now, combine the results:

cos⁡(−60°) + sin⁡(−30°) = 1/2 + (-1/2) = 1/2 - 1/2 = 0

Question 2: Prove that sin⁡(−x) + sin⁡(x) = 0

Solution:

Using the odd identity for sine:
sin⁡(−x) = −sin⁡(x)

Now substitute this into the left side of the equation:
sin⁡(−x) + sin⁡(x) = −sin⁡(x) + sin⁡(x) = 0

Thus, the identity is proven to be true.

Question 3: Find cos⁡(−135°) and sin⁡(−135°).

Solution:

• For cos⁡(−135°): cos⁡(−135°) = cos⁡(135°) = cos⁡(90° + 45°) = - sin(45°) = −1/√2​​ [As cos(90° + x) = -sin x]
• For sin⁡(−135°): sin⁡(−135°) = − sin⁡(135°) = -sin(90° + 45°) = -cos(45°) = −1/√2 [As sin(90° + x) = cos x]

Worksheet: Even-Odd Identities

You can download this free worksheet on Even-Odd Identities from below:

Conclusion

In conclusion, even-odd identities are important concepts in trigonometry that help us understand how sine and cosine functions behave with negative angles. By recognizing that cosine is an even function and sine is an odd function, we can simplify calculations and solve problems more easily.

Read More,

• Trigonometric Identities
• Trigonometry Formulas
• Inverse Trigonometric Identities