Trigonometric Functions: Detailed Explanation, Proofs, and Derivations
Trigonometric functions are fundamental in the study of geometry, especially in the context of right-angled triangles and the unit circle. The primary trigonometric functions are sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). These functions are defined based on the relationships between the angles and sides of a right triangle, and they also have important representations on the unit circle.
Definitions
For a right-angled triangle with an angle \(\theta\), the definitions of the primary trigonometric functions are as follows:
$$ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} $$
$$ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} $$
$$ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\text{opposite}}{\text{adjacent}} $$
The Unit Circle
The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system. The angle \(\theta\) is measured from the positive x-axis. A point on the unit circle corresponding to the angle \(\theta\) has coordinates \((\cos \theta, \sin \theta)\).
Pythagorean Identity
One of the fundamental identities in trigonometry is the Pythagorean identity, which states that for any angle \(\theta\):
$$ \sin^2 \theta + \cos^2 \theta = 1 $$
This can be derived from the Pythagorean theorem applied to the unit circle.
Derivatives and Integrals
The derivatives and integrals of trigonometric functions are essential in calculus. The basic derivatives are:
$$ \frac{d}{d\theta} (\sin \theta) = \cos \theta $$
$$ \frac{d}{d\theta} (\cos \theta) = -\sin \theta $$
$$ \frac{d}{d\theta} (\tan \theta) = \sec^2 \theta $$
The basic integrals are:
$$ \int \sin \theta \, d\theta = -\cos \theta + C $$
$$ \int \cos \theta \, d\theta = \sin \theta + C $$
$$ \int \tan \theta \, d\theta = -\ln |\cos \theta| + C $$
Addition and Subtraction Formulas
The addition and subtraction formulas for sine and cosine are as follows:
$$ \sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta $$
$$ \sin (\alpha – \beta) = \sin \alpha \cos \beta – \cos \alpha \sin \beta $$
$$ \cos (\alpha + \beta) = \cos \alpha \cos \beta – \sin \alpha \sin \beta $$
$$ \cos (\alpha – \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta $$
Proof of the Pythagorean Identity
To prove the Pythagorean identity, consider a point on the unit circle with coordinates \((x, y)\). By definition, \(x = \cos \theta\) and \(y = \sin \theta\). The equation of the unit circle is:
$$ x^2 + y^2 = 1 $$
Substituting the trigonometric functions:
$$ \cos^2 \theta + \sin^2 \theta = 1 $$
This confirms the Pythagorean identity.
Euler’s Formula
Euler’s formula is a profound equation in mathematics that establishes the fundamental relationship between trigonometric functions and the exponential function. It states that for any real number \(\theta\):
$$ e^{i\theta} = \cos \theta + i \sin \theta $$
This can be derived from the Taylor series expansions of \(e^{i\theta}\), \(\cos \theta\), and \(\sin \theta\).
Proof of Euler’s Formula
Starting with the Taylor series expansion for the exponential function:
$$ e^{i\theta} = \sum_{n=0}^{\infty} \frac{(i\theta)^n}{n!} $$
Separating the series into real and imaginary parts:
$$ e^{i\theta} = \sum_{k=0}^{\infty} \frac{(-1)^k \theta^{2k}}{(2k)!} + i \sum_{k=0}^{\infty} \frac{(-1)^k \theta^{2k+1}}{(2k+1)!} $$
Recognizing the Taylor series for cosine and sine:
$$ e^{i\theta} = \cos \theta + i \sin \theta $$
This completes the proof of Euler’s formula.
