Trig Cheat Sheet Notes

Definition of the Trig Functions

Right Triangle Definition

• Assumes: (0 < \theta < \pi/2) or (0 < \theta < 90^\circ).
• Definitions:
  • ( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} )
  • ( \csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} )
  • ( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} )
  • ( \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} )
  • ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} )
  • ( \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} )

Unit Circle Definition

• ( \theta ) is any angle.
• Definitions:
  • ( \sin(\theta) = y )
  • ( \csc(\theta) = \frac{1}{y} )
  • ( \cos(\theta) = x )
  • ( \sec(\theta) = \frac{1}{x} )
  • ( \tan(\theta) = \frac{y}{x} )
  • ( \cot(\theta) = \frac{x}{y} )

Facts and Properties

Domain

• ( \sin(\theta) ) and ( \cos(\theta) ): ( \theta ) can be any angle.
• ( \tan(\theta) ) and ( \sec(\theta) ): Undefined for ( \theta = \frac{\pi}{2} + n\pi ), ( n = 0, 1, 2, \ldots )
• ( \csc(\theta) ) and ( \cot(\theta) ): Undefined for ( \theta = n\pi ), ( n = 0, 1, 2, \ldots )

Period

• ( \sin(\theta), \cos(\theta), \csc(\theta), \sec(\theta) ): Period = (2\pi)
• ( \tan(\theta), \cot(\theta) ): Period = (\pi)

Range

• ( -1 \leq \sin(\theta) \leq 1
• ( -1 \leq \cos(\theta) \leq 1
• ( -\infty < \tan(\theta) < \infty
• ( -\infty < \cot(\theta) < \infty
• ( \sec(\theta) \geq 1 \text{ or } \sec(\theta) \leq -1
• ( \csc(\theta) \geq 1 \text{ or } \csc(\theta) \leq -1

Formulas and Identities

Tangent and Cotangent Identities

• ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )
• ( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} )

Reciprocal Identities

• ( \csc(\theta) = \frac{1}{\sin(\theta)} )
• ( \sec(\theta) = \frac{1}{\cos(\theta)} )
• ( \cot(\theta) = \frac{1}{\tan(\theta)} )

Pythagorean Identities

• ( \sin^2(\theta) + \cos^2(\theta) = 1 )
• ( \tan^2(\theta) + 1 = \sec^2(\theta) )
• ( 1 + \cot^2(\theta) = \csc^2(\theta) )

Even/Odd Formulas

• Even: ( \cos(-\theta) = \cos(\theta), \sec(-\theta) = \sec(\theta) )
• Odd: ( \sin(-\theta) = -\sin(\theta), \csc(-\theta) = -\csc(\theta) )
• ( \tan(-\theta) = -\tan(\theta), \cot(-\theta) = -\cot(\theta) )

Periodic Formulas

• ( \sin(\theta + 2\pi n) = \sin(\theta) )
• ( \cos(\theta + 2\pi n) = \cos(\theta) )
• ( \tan(\theta + \pi n) = \tan(\theta) )

Degrees to Radians Conversion

• ( 180^\circ = \pi \text{ radians} )
• ( t = \frac{x \pi}{180} )
• ( x = \frac{180t}{\pi} )

Double Angle Formulas

• ( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) )
• ( \cos(2\theta) = \cos^2(\theta) - \sin^2(\theta) )
• ( \tan(2\theta) = \frac{2 \tan(\theta)}{1 - \tan^2(\theta)} )

Half Angle Formulas

• ( \sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos(\theta)}{2}} )
• ( \cos(\frac{\theta}{2}) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}} )
• ( \tan(\frac{\theta}{2}) = \frac{\sqrt{1 - \cos(\theta)}}{\sqrt{1 + \cos(\theta)}} )

Sum and Difference Formulas

• ( \sin(\alpha \pm \beta) = \sin(\alpha) \cos(\beta) \pm \cos(\alpha) \sin(\beta) )
• ( \cos(\alpha \pm \beta) = \cos(\alpha) \cos(\beta) \mp \sin(\alpha) \sin(\beta) )
• ( \tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)} )

Product to Sum Formulas

• ( \sin(A) \sin(B) = \frac{1}{2} [\cos(A - B) - \cos(A + B)] )
• ( \cos(A) \cos(B) = \frac{1}{2} [\cos(A - B) + \cos(A + B)] )

Sum to Product Formulas

• ( \sin(A) + \sin(B) = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) )
• ( \cos(A) + \cos(B) = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) )

Cofunction Formulas

• ( \sin\left(\frac{\pi}{2} - \theta\right) = \cos(\theta) )
• ( \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) )

Inverse Trig Functions

Definition

• ( y = \sin^{-1}(x) \Leftrightarrow x = \sin(y) )
• ( y = \cos^{-1}(x) \Leftrightarrow x = \cos(y) )
• ( y = \tan^{-1}(x) \Leftrightarrow x = \tan(y) )

Domain and Range

• ( y = \sin^{-1}(x) ): Domain ([-1, 1]), Range ([-\frac{\pi}{2}, \frac{\pi}{2}])
• ( y = \cos^{-1}(x) ): Domain ([-1, 1]), Range ([0, \pi])
• ( y = \tan^{-1}(x) ): Domain ((-\infty, \infty)), Range ((-\frac{\pi}{2}, \frac{\pi}{2}))

Inverse Properties

• ( \cos(\cos^{-1}(x)) = x )
• ( \sin(\sin^{-1}(x)) = x )
• ( \tan(\tan^{-1}(x)) = x )

Alternate Notation

• ( \sin^{-1}(x) = \text{arcsin}(x) )
• ( \cos^{-1}(x) = \text{arccos}(x) )
• ( \tan^{-1}(x) = \text{arctan}(x) )

Law of Sines, Cosines, and Tangents

Law of Sines

• ( \frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c} )

Law of Cosines

• ( a^2 = b^2 + c^2 - 2bc \cos(A) )

Mollweide's Formula

• ( \frac{a + b}{c} = \frac{\cos\left(\frac{A - B}{2}\right)}{\sin\left(\frac{C}{2}\right)} )

Law of Tangents

• ( \frac{a - b}{a + b} = \tan\left(\frac{A - B}{2}\right) \cot\left(\frac{A + B}{2}\right) )

Note: These notes summarize the key elements of trigonometric identities and functions useful for mathematical problems, derived from Paul Dawkins' Trig Cheat Sheet available at tutorial.math.lamar.edu.