Trigonometric identities cover a wide range, and many people find memorising them challenging. Much like mnemonic devices used to remember historical dates, you might want to memorise these identities quickly. While rote memorisation is not without merit in mathematics, relying solely on it offers limited benefits.
By proving trigonometric identities, you gain insight into crucial algebraic manipulations and ways of thinking within this field. Moreover, understanding how identities are derived means you do not need to memorise the numerous trigonometric identities.
This article highlights the importance of proving identities, focusing on what you gain beyond simple memorisation.
1 Deepening Understanding of Definitions and Concepts of Trigonometric identities
In mathematics, not only in trigonometry, it is common to start proofs of identities and theorems from definitions. Regular problem-solving may not frequently require direct use of definitions, causing many to forget them over time.
Even if a property might seem obvious with a solid understanding of definitions, a lack of comprehension can prevent you from recognising these features, hindering your overall grasp of mathematics. For instance, the definition of trigonometric functions is given using the unit circle, but it can also be described with a circle of any positive radius r. Is this acceptable—do the values depend on the circle’s radius?
It turns out that trigonometric functions do not depend on the radius. Whether the radius is 1 or 2, the values for the same angle remain unchanged. Comparing the unit circle and a circle scaled by r, the trigonometric values for a given angle are identical (Figure 1). Since r can be any positive real number, we conclude that trigonometric values are independent of the circle’s radius.
Therefore, it makes no difference whether trigonometric functions are defined on a circle with radius 1 or radius r.
It is common to learn a definition once and then barely use it, eventually forgetting it. However, by proving theorems and deepening your understanding of definitions, you develop a clearer knowledge of properties such as those of trigonometric functions.
2 Important Algebraic Manipulations and Ways of Thinking in Trigonometry
By proving trigonometric identities, you learn essential algebraic transformations. Consider the double-angle identity:
Double-Angle Identity
\sin{2\theta} = 2\sin\theta \cos\theta \tag{1}
This identity can be easily proved using the addition formulae.
Starting from the addition formula:
\sin(A + B) = \sin A \cos B + \cos A \sin B
Put A = \theta and B = \theta to obtain:
\begin{aligned} \sin(\theta + \theta) &= \sin \theta \cos \theta + \cos \theta \sin \theta\\ &= 2 \sin \theta \cos \theta. \end{aligned}
This completes the proof. The addition formula allows us to break down the trigonometric function in A + B into the sum of the functions in A and B. When A and B are equal, the identity becomes the double-angle identity.
Now, let us consider how to transform the expression:
\sin(3 \theta)
This appears in the triple-angle identity and can be transformed similarly. As explained in the proof of the double-angle identity, the addition formulae breaks down an angle A + B into A and B. Since 3 \theta = \theta + 2 \theta, we can write:
\begin{aligned} \sin(3 \theta) &= \sin (\theta + 2 \theta)\\ &= \sin \theta \cos 2 \theta + \cos \theta \sin 2 \theta. \end{aligned}
Both \cos 2 \theta and \sin 2 \theta appear here, so let us apply the double-angle identities:
\begin{aligned} \sin \theta \cos 2 \theta + \cos \theta \sin 2 \theta &= \sin \theta (1 - 2\sin^2\theta) + \cos \theta (2 \sin \theta \cos \theta) \qquad (\because \text{double-angle identities})\\ &= \sin \theta - 2 \sin^3 \theta + 2 \sin \theta \cos^2 \theta\\ &= \sin \theta - 2 \sin^3 \theta + 2 \sin \theta (1 - \sin^2 \theta) \quad (\because \sin^2 \theta + \cos^2 \theta = 1)\\ &= 3 \sin \theta - 4 \sin^3 \theta. \end{aligned}
Hence, we obtain:
\sin(3 \theta) = 3 \sin \theta - 4 \sin^3 \theta.
Memorising identities certainly has value, but more important is understanding why we applied the addition formula. Here, we used its property to break the angle A + B into A and B.
By proving trigonometric identities, you learn typical algebraic transformations and the ideas behind proofs, which cultivates your ability to apply these concepts flexibly.
3 Intuitive Understanding of Trigonometric Functions
\sin \theta and \cos \theta only take values between -1 and 1, as expressed by:
\begin{aligned} -1 &\le \sin \theta \le 1,\\ -1 &\le \cos \theta \le 1.\\ \end{aligned} \tag{2}
Why do trigonometric functions take values in this range? As discussed in Section 1, proving trigonometric identities deepens your understanding of their definitions. This deeper understanding makes properties like those in Equation 2 intuitive.
Consequently, you reduce careless mistakes such as forgetting conditions or overlooking unreasonable results.
Using the definition \sin \theta = \frac{y}{r} and \cos \theta = \frac{x}{r}, where (x,y) is a point on a circle of radius r, you immediately see that y and x can only take values between -r and r. Thus, Equation 2 naturally holds.
If you are familiar with the definition of trigonometric functions, this kind of reasoning becomes second nature. Therefore, if you encounter a value like \sin \theta = 2, you quickly recognise that this is not possible. Although memorising Equation 2 helps you notice these mistakes, there are many properties of trigonometric functions besides this. Solely relying on memorisation increases the chance of forgetting crucial conditions.
4 Reducing the Number of Trigonometric Identities You Need to Memorise
There are many trigonometric identities, but understanding how to prove them means you need to memorise far fewer.
I classify trigonometric identities into two groups:
- Those proved using the definition of trigonometric functions.
- Those proved using the addition formulae.
In other words, if you memorise the definition and the addition formulae, you do not have to memorise all other identities. The definition is fundamental and should be well understood and remembered. The addition formulae must be memorised for exams since they are somewhat tricky to prove,
However, once you are accustomed to prove identities, you greatly reduce the number of proofs you need to memorise. Given that the proofs in trigonometry are relatively straightforward, you can even quickly derive identities during exams, which is highly practical. This advantage stems from understanding identities rather than rote learning.
5 How to Become Comfortable with Proofs of Trigonometric Identities
Proving identities does not naturally become part of your knowledge after just one attempt. Practice deriving identities repeatedly.
What matters most during these derivations is to “organise in your own mind why you thought about things this way.” Always reflect on “what you need to know to be able to derive it yourself next time.” Unlike rote memorisation, once you understand the reasoning behind a method, it tends to be retained.
Alongside this, solve problems using these identities repeatedly. It is acceptable to refer to the identities while solving. Frequent use naturally reinforces your recall. You may also want to derive the identities again when you forget them. This way, you can quickly reproduce identities even under exam conditions.
6 Conclusion
Though proving identities may seem a roundabout way, if you seriously aim to improve your mathematical skills, it is unavoidable. Compared to simple memorisation, proving identities teaches you the fundamental concepts and common algebraic transformations essential for applications.
Focus on understanding derivations and practise problem-solving to naturally become capable of proving identities yourself.
