What Techniques Can Be Used to Teach Yourself Trigonometry?

It's often not as challenging as it might seem at first glance. Trigonometry is simply about ratios related to three key elements: the sine, cosine, and tangent. In a right-angled triangle, for instance, two angles (other than the 90-degree angle) and three sides are involved. This reduces to two variables to focus on at a time.

The 90-degree angle is fixed, and the hypotenuse, the triangle's longest side, is opposite to it, which leaves two sides of varying lengths to consider. These sides change depending on the angles opposite them. By comparing these sides, we derive the sine, cosine, and tangent ratios.

Simplifying Trigonometry with Basic Ratios

There are only three primary ratios to remember: sine (sin), cosine (cos), and tangent (tan). In a right-angled triangle, the sine of an angle is the ratio of the opposite side to the hypotenuse, cosine is the ratio of the adjacent side to the hypotenuse, and tangent is the ratio of the opposite side to the adjacent side. These can be expressed as:

sin θ opposite side / hypotenuse cos θ adjacent side / hypotenuse tan θ opposite side / adjacent side sin θ / cos θ

Everything else in trigonometry can essentially be derived from these three ratios. For instance, secant (sec), cosecant (csc), and cotangent (cot) are just the reciprocal versions of sec, csc, and cot respectively. Knowing these, you can already work with a wide range of trigonometric problems.

For values of specific angles, such as 0, 30, 45, 60, and 90 degrees, and their multiples, you can either memorize them or use tables and calculators to find them quickly. However, if you need to work with angles that are not these standard ones, you would either need to be able to draw and measure the lengths of the triangle or derive the values yourself.

Challenges Faced by Students

Many students find trigonometry challenging because it is often abstract and disconnected from real-life applications. It is often viewed merely as an academic subject with no practical relevance, leading to a lack of interest and engagement. However, trigonometry is essential in various fields, such as physics, engineering, and navigation, making it a crucial skill to master.

The Harvard University Department of Mathematics mentions that Math 55 is a notoriously difficult course that even some of the brightest students struggle with. This highlights the difficulty but also underscores the importance of a solid foundation. Understanding and applying trigonometry principles can be challenging, and without a strong grounding, students may find it difficult to grasp the more advanced concepts.

Effective Strategies for Learning Trigonometry

One way to overcome the challenge is to use a variety of resources. A few effective strategies include:

Online Learning: Watching lectures from reputable sources like Professor Leonard's YouTube channel, who offers detailed explanations for Calculus I-III, can be immensely helpful. While these lectures are long, they are thorough and insightful. If you're particularly interested in trigonometry, you might want to find resources that focus specifically on this topic, such as Khan Academy's Trigonometry section. Books: A good book on trigonometry can provide structured guidance and detailed explanations. Make sure the book covers the basics thoroughly and provides plenty of examples and exercises. Seeking Help: Sometimes, the most effective way to learn is by seeking help from a mentor, a colleague, or a teacher who can provide personalized guidance and feedback. Engaging in a discussion or a problem-solving session can clarify doubts that written materials might not address effectively.

The Role of Visual Aids

A unit circle is a valuable tool in visualizing trigonometric functions. For trigonometry, it's useful to start with a right-angled triangle drawn on the radius of a unit circle. Here, θ is the angle at the center of the circle, with 'o' as the vertical side (opposite angle θ) and 'a' as the horizontal side (adjacent to angle θ). The hypotenuse, h, is the radius and equals 1. Using these, you can easily understand the basic trigonometric ratios:

RatioDescriptionEquation sin θOpposite side / Hypotenuseo/h cos θAdjacent side / Hypotenusea/h tan θOpposite side / Adjacent sideo/a sin θ / cos θ

Note: The equation sin θ / cos θ o/h / a/h oh / ah a/h tan θ shows how the tangent is derived from sine and cosine.

Conclusion

Trigonometry is fundamentally about understanding ratios and their relationships in right-angled triangles. With the right approach, it can be a straightforward and rewarding subject to learn. Whether you're learning trigonometry for academic purposes or practical applications, having a solid grasp of the basics and utilizing helpful resources will make the journey smoother.

Key Takeaways:

Trigonometry basics involve understanding sine, cosine, and tangent ratios. Using resources like Khan Academy, Professor Leonard's lectures, and textbooks can be very helpful. Visual aids like the unit circle and real-life applications can make trigonometry more relatable and easier to understand.

Embrace the challenge and dive into the wonderful world of trigonometry!