AcademyTrigonometry
Academy
Trig Functions
Level 1 - Math I (Physics) topic page in Trigonometry.
Trigonometric Functions
The three primary trigonometric functions relate the angles of a right triangle to the ratios of its sides.
Definitions
Given a right triangle with angle \(\theta\), we define:
Sine
\[\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\]
Cosine
\[\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\]
Tangent
\[\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}\]
The Unit Circle
The unit circle is a circle with radius 1 centered at the origin. For any angle \(\theta\), the coordinates of the point where the terminal side intersects the unit circle are \((\cos\theta, \sin\theta)\).
Key angles and their values:
- \(\sin(0) = 0\), \(\cos(0) = 1\), \(\tan(0) = 0\)
- \(\sin\left(\frac{\pi}{2}\right) = 1\), \(\cos\left(\frac{\pi}{2}\right) = 0\), \(\tan\left(\frac{\pi}{2}\right)\) is undefined
- \(\sin(\pi) = 0\), \(\cos(\pi) = -1\), \(\tan(\pi) = 0\)
Graphs and Periodicity
The trigonometric functions are periodic, meaning they repeat their values in a regular cycle.
- Sine and cosine have period \(2\pi\): \(\sin(\theta + 2\pi) = \sin\theta\), \(\cos(\theta + 2\pi) = \cos\theta\)
- Tangent has period \(\pi\): \(\tan(\theta + \pi) = \tan\theta\)
The amplitude of sine and cosine is 1, while tangent has no bounded amplitude.
Reciprocal Functions
Three additional trigonometric functions exist as reciprocals:
Cosecant
\[\csc\theta = \frac{1}{\sin\theta}\]
Secant
\[\sec\theta = \frac{1}{\cos\theta}\]
Cotangent
\[\cot\theta = \frac{1}{\tan\theta} = \frac{\cos\theta}{\sin\theta}\]