Even And Odd Trigonometric Functions

What Are Even and Odd Functions?

An even trigonometric function is a function that is symmetric with respect to the y-axis, while an odd trigonometric function is symmetric with respect to the origin. Before we apply this concept to trigonometry, let's refresh our memory of the algebraic definitions of even and odd functions. These definitions are the foundation for everything we will explore.

A function, let's call it $f(x)$, is defined as even if, for every value of $x$ in its domain, the following equation holds true:

What does this mean? It means that if you plug in a negative input ($-x$), you get the exact same output as if you plugged in the positive version of that input ($x$). The classic example is the parabola $f(x)=x^2$. If you calculate $f(2)$, you get $2^2=4$. If you calculate $f(-2)$, you get $(-2{)}^2=4$. The outputs are identical. Graphically, this property creates perfect symmetry across the y-axis. If you were to fold the graph along the y-axis, the left and right sides would match up perfectly.

On the other hand, a function $f(x)$ is defined as odd if, for every value of $x$ in its domain, this equation is true:

This definition tells us that plugging in a negative input gives you the negative of the output for the positive input. Consider the function $f(x)=x^3$. We find that $f(2)=2^3=8$. For the opposite input, $f(-2)=(-2{)}^3=-8$. Notice that $f(-2)$ is the exact negative of $f(2)$. This relationship results in what is called origin symmetry. If you rotate the graph ${180}^{\circ }$ around the origin (the point $(0,0)$), it will land back on top of itself.

Now, our goal is to apply these same algebraic tests to the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent. We will substitute a negative angle, like $-\theta$, into each function and observe whether the result is the original function (even), the negative of the original function (odd), or something else entirely (neither).

How Do We Test Trigonometric Functions for Symmetry?

The key to determining whether a trigonometric function is even or odd lies in understanding the geometry of the unit circle. The unit circle is a circle with a radius of $1$ centered at the origin of the Cartesian plane. Any point $P$ on the circle can be described by coordinates $(x,y)$, where $x=\cos (\theta )$ and $y=\sin (\theta )$, and $\theta$ is the angle measured from the positive x-axis.

The test for symmetry involves comparing an angle $\theta$ with its negative counterpart, $-\theta$. Let's visualize this:

• A positive angle $\theta$ is formed by rotating counter-clockwise from the positive x-axis.
• A negative angle $-\theta$ is formed by rotating clockwise from the positive x-axis by the same magnitude.

Imagine a point $P(x,y)$ on the unit circle that corresponds to a positive angle $\theta$. Now, consider the point $P^{\prime }$ that corresponds to the angle $-\theta$. Because we are just reflecting the angle across the x-axis, the x-coordinate of $P^{\prime }$ will be the same as the x-coordinate of $P$. However, the y-coordinate of $P^{\prime }$ will be the negative of the y-coordinate of $P$. Therefore, the coordinates of $P^{\prime }$ are $(x,-y)$.

This single geometric insight is incredibly powerful. Let's summarize the relationships:

• For angle $\theta$: Point is $(x,y)$, so $\cos (\theta )=x$ and $\sin (\theta )=y$.
• For angle $-\theta$: Point is $(x,-y)$, so $\cos (-\theta )=x$ and $\sin (-\theta )=-y$.

By comparing these results, we can directly apply the algebraic definitions of even and odd functions. We will see that $\cos (-\theta )$ is identical to $\cos (\theta )$, making cosine an even function. In contrast, $\sin (-\theta )$ is the negative of $\sin (\theta )$, making sine an odd function. These two fundamental results will allow us to classify all six trigonometric functions.

The Even Trigonometric Functions: Cosine and Secant

As we established using the unit circle, the x-coordinate for an angle $\theta$ and an angle $-\theta$ is identical. Since the cosine function is defined as the x-coordinate on the unit circle, we arrive at a crucial identity.

For any angle $\theta$, we have $\cos (\theta )=x$ and $\cos (-\theta )=x$. Comparing these, we see that:

This equation perfectly matches the definition of an even function, $f(-x)=f(x)$. Therefore, cosine is an even function. The graph of $y=\cos (x)$ is a clear illustration of this property; it is perfectly symmetric with respect to the y-axis.

What about the reciprocal of cosine, the secant function? We can use the identity for cosine to determine its nature. Recall that $\sec (\theta )=\frac{1}{\cos (\theta )}$. Let's test it by substituting $-\theta$:

$\sec (-\theta )=\frac{1}{\cos (-\theta )}$

Since we know $\cos (-\theta )=\cos (\theta )$, we can substitute this into the equation:

$\sec (-\theta )=\frac{1}{\cos (\theta )}$

And since $\frac{1}{\cos (\theta )}=\sec (\theta )$, we have our result:

This shows that the secant function is also an even function. It inherits its symmetry from the cosine function. In summary, of the six trigonometric functions, only cosine and secant are even.

The Odd Trigonometric Functions: Sine, Cosecant, Tangent, and Cotangent

Now let's turn our attention to the remaining four trigonometric functions. From our unit circle analysis, we saw that the y-coordinate for an angle $-\theta$ is the negative of the y-coordinate for $\theta$. Since the sine function is defined as the y-coordinate, this leads directly to its classification.

For any angle $\theta$, $\sin (\theta )=y$ and $\sin (-\theta )=-y$. By substituting $\sin (\theta )$ for $y$, we get:

This identity matches the definition of an odd function, $f(-x)=-f(x)$. Therefore, sine is an odd function. Its graph displays symmetry about the origin; a ${180}^{\circ }$ rotation leaves the graph unchanged.

We can use this primary result, along with the fact that cosine is even, to determine the nature of the other three functions.

• Cosecant (csc): As the reciprocal of sine, $\csc (\theta )=\frac{1}{\sin (\theta )}$. Let's test $-\theta$:
  $\csc (-\theta )=\frac{1}{\sin (-\theta )}=\frac{1}{-\sin (\theta )}=-\frac{1}{\sin (\theta )}=-\csc (\theta )$. Cosecant is odd.
$\csc (-\theta )=-\csc (\theta )$
• Tangent (tan): As the ratio of sine to cosine, $\tan (\theta )=\frac{\sin (\theta )}{\cos (\theta )}$. Let's test $-\theta$:
  $\tan (-\theta )=\frac{\sin (-\theta )}{\cos (-\theta )}=\frac{-\sin (\theta )}{\cos (\theta )}=-\tan (\theta )$. Tangent is odd.
$\tan (-\theta )=-\tan (\theta )$
• Cotangent (cot): As the ratio of cosine to sine, $\cot (\theta )=\frac{\cos (\theta )}{\sin (\theta )}$. Let's test $-\theta$:
  $\cot (-\theta )=\frac{\cos (-\theta )}{\sin (-\theta )}=\frac{\cos (\theta )}{-\sin (\theta )}=-\cot (\theta )$. Cotangent is odd.
$\cot (-\theta )=-\cot (\theta )$

So, sine and its reciprocal (cosecant), as well as tangent and its reciprocal (cotangent), are all odd functions.

How Are These Properties Used to Simplify Expressions?

The even-odd properties, also known as the negative angle identities, are powerful tools in trigonometry. Their primary application is to simplify expressions, especially those containing negative angles. By applying these identities, you can often transform a complicated-looking expression into a much simpler, more manageable form. This skill is crucial for solving trigonometric equations and proving more complex identities.

The general strategy is to scan an expression for any trigonometric function with a negative angle as its argument, such as $\sin (-x)$ or $\cos (-3\theta )$. When you find one, replace it with its equivalent form based on its even or odd property. For even functions (cos, sec), you can simply remove the negative sign. For odd functions (sin, csc, tan, cot), you remove the negative sign from the angle and place it in front of the entire function.

Common Mistakes to Avoid

When working with even and odd trigonometric functions, a few common pitfalls can lead to incorrect answers. Being aware of these can help you build confidence and accuracy.

• Mixing Up the Functions: The most frequent error is simply misremembering which functions are even and which are odd. Many students incorrectly assume sine is even. A good mnemonic is that Cosine and its reCiprocal, seCant, are the only even functions.
• Sign Errors: Forgetting to place the negative sign in front of an odd function is a critical mistake. For example, writing $\tan (-x)=\tan (x)$ instead of $\tan (-x)=-\tan (x)$ will completely change the outcome of a simplification or proof. Always double-check your signs.
• Incorrectly Handling Binomials: The even-odd properties apply to arguments like $(-x)$, not binomials like $(x-\pi )$. A common mistake is to think $\cos (x-\pi )=\cos (x)-\cos (\pi )$ or $\cos (\pi -x)=\cos (x)$. This is incorrect. You must use sum and difference formulas for such expressions. The negative angle identity is a specific case.
• Confusing Even/Odd with Cofunctions: The even-odd identities relate $f(x)$ and $f(-x)$. Cofunction identities relate a function to its "co-" function, like $\sin (x)=\cos (\frac{\pi }{2}-x)$. These are different sets of tools for different situations. Don't mix them up.
• Assuming a Shifted Function is Still Even or Odd: The function $f(x)=\cos (x)$ is even. However, a horizontally shifted version, like $g(x)=\cos (x-1)$, is neither even nor odd. The symmetry is lost once the graph is shifted away from the y-axis (for even) or the origin (for odd).

Quick Summary and Reference Table

The concepts of even and odd functions provide a powerful way to understand the inherent symmetries of the trigonometric functions. An even function, like cosine, has y-axis symmetry, meaning $f(-x)=f(x)$. An odd function, like sine, has origin symmetry, meaning $f(-x)=-f(x)$. These properties, also called negative angle identities, are essential for simplifying expressions and proving other identities in trigonometry and calculus.

Here is a comprehensive table to use as a quick reference: