When students learn about trigonometric functions, they often make some common mistakes that can hurt their understanding and grades.

Making Mistakes with Derivatives
One big mistake is forgetting the correct derivatives of trigonometric functions. For example, students might remember that the derivative of $\sin(x)$ is $\cos(x)$, but they might get it mixed up and say the opposite. It's really important to remember these derivatives correctly:

• The derivative of $\sin(x)$ is $\cos(x)$
• The derivative of $\cos(x)$ is $-\sin(x)$
• The derivative of $\tan(x)$ is $\sec^2(x)$

Forgetting the Chain Rule
Another mistake is when students forget to use the chain rule while working on functions that are combined together. For example, if you need to find the derivative of $\sin(3x)$, students might forget to multiply by the derivative of the inside function (which is $3x$). To do it right, you would say:
$\text{The derivative of } \sin(3x) \text{ is } 3\cos(3x)$

Using Identities Incorrectly
Sometimes, students also mix up trigonometric identities. This can make problems harder than they need to be and lead to wrong answers. This is especially true when dealing with products (like multiplying) or quotients (like dividing) of trigonometric functions. Knowing how to use the right identities is very important for getting the right derivatives.

Not Thinking About the Unit Circle
Lastly, when students are solving problems about trigonometric derivatives, some forget to think about the unit circle. This can lead to mistakes in understanding how these functions behave or repeat over time. Getting comfortable with the unit circle can really help improve understanding and accuracy in differentiation.