The connections between the sine, cosine, and tangent graphs can be tricky for 12th-grade students, especially those studying under the British AS-Level curriculum. To really understand these relationships, it’s important to know what these functions mean and how they look on a graph.

1. Sine and Cosine Functions:

• The sine function, written as ( y = \sin(x) ), and the cosine function, written as ( y = \cos(x) ), both have a pattern that repeats every ( 2\pi ).
• Their values go up and down between -1 and 1. This can confuse students because not all graphs behave the same way. Also, people often get mixed up about how these functions relate to one another. The cosine graph is actually ahead of the sine graph by ( \frac{\pi}{2} ).
• The height (amplitude) and how quickly they repeat (period) are important features, but these can make it harder to understand. For example, students might think there are differences in how these two graphs repeat or in their heights, which isn’t true.

2. Tangent Function:

• The tangent function, shown as ( y = \tan(x) ), doesn’t repeat in the same way as sine and cosine. It has some unique traits, like vertical lines where it can't be defined, specifically at ( x = \frac{\pi}{2} + n\pi ) (where ( n ) is any whole number).
• This adds some difficulty because students need to deal with these breaks in the graph while remembering that tangent does repeat every ( \pi ). If students don’t understand where these lines are, they can make big mistakes when trying to draw or read the tangent graph.

3. Interrelationships:

• The link between these functions gets even more complicated when students look at the relationship ( y = \tan(x) = \frac{y = \sin(x)}{y = \cos(x)} ). This idea is really important, but it’s often hard for students to realize how the sine and cosine graphs affect the tangent graph.
• So, the graphs don’t just share similarities in height and repeating patterns; they also define each other. Not understanding how they connect can make learning about trigonometric identities hard and their uses unclear.

4. Resolving the Difficulties:

• One great way to tackle these challenges is by using interactive graphing tools or graphing calculators. These tools let students see the functions change in real-time. Watching how changing values affects sine and cosine can help clarify their relationships.
• Also, comparing the graphs by putting them on the same set of axes can help students visually see how they connect and how they are different.
• Breaking down sine and cosine functions into simpler parts can sometimes help, but it could also make things more complicated for some students.

In conclusion, while the relationship between sine, cosine, and tangent functions can make studying trigonometry tricky, using technology, working together with classmates, and careful analysis can help reduce confusion and improve understanding.