Solutions to Questions on Trigonometric Functions

Here are given the solutions and answers to the questions in Graphs of Basic Trigonometric Functions. For each trigonometric function is given the domain, range, period and asymptotes (if any). These properties are necessary to understand the graphs of the trigonometric functions. The relationship between trigonometric functions are also explored.

TUTORIAL (1) - Domain, Range, Zeros and Vertical Asymptotes of the 6 Basic Trigonometric Functions - Answers

1. \( f(x) = \sin(x) \)
   • domain : \( (-\infty , +\infty) \)
   • range : \( [-1 , 1] \)
   • period : \( 2\pi \)
   • zeros at \( x = k\pi, k \text{ is an integer } : k = 0 , \pm 1 , \pm 2 , ... \)
   
   
   
2. \( f(x) = \cos(x) \)
   • domain : \( (-\infty , +\infty) \)
   • range : \( [-1 , 1] \)
   • period : \( 2\pi \)
   • zeros at \( x = \dfrac{\pi}{2} + k\pi, k \text{ is an integer } : k = 0 , \pm 1 , \pm 2 , ... \)
   
   
   
3. \( f(x) = \tan(x) \)
   • domain : all real numbers except \( \dfrac{\pi}{2} + k\pi \) where \( k \) is an integer.
   • range : \( (-\infty , +\infty) \)
   • period : \( \pi \)
   • zeros at \( x = k\pi, k \text{ is an integer } : k = 0 , \pm 1 , \pm 2 , ... \). (Note: \( \tan(x) \) and \( \sin(x) \) have the same zeros because \( \tan(x) = \dfrac{\sin(x)}{\cos(x)} \)
   • vertical asymptotes at \( x = \dfrac{\pi}{2} + k\pi\) wnere \( k \) is an integer.
   
   
   
4. \( f(x) = \cot(x) \)
   • domain : all real numbers except \( k\pi \) where \( k \) is an integer.
   • range : \( (-\infty , +\infty) \)
   • period : \( \pi \)
   • zeros at \( x = \dfrac{\pi}{2} + k\pi, k \text{ is an integer } : k = 0 , \pm 1 , \pm 2 , ...\). (Note: \( \cot(x) \) and \( \cos(x) \) have the same zeros because \( \cot(x) = \dfrac{\cos(x)}{\sin(x)} \)
   • vertical asymptotes at \( x = k\pi \) where \( k \) is an integer.
   
   
   
5. \( f(x) = \sec(x) \)
   • domain : all real numbers except \( \dfrac{\pi}{2} + k\pi \) where \( k \) is an integer.
   • range : \( (-\infty , -1] \cup [1 , +\infty) \)
   • period : \( 2\pi \)
   • \( \sec(x) \) has no zeros.
   • vertical asymptotes at \( x = \pi/2 + k\pi \) k is an integer.
   
   
   
6. \( f(x) = \csc(x) \)
   • domain : all real numbers except \( k\pi \) where \( k \) is an integer.
   • range : \( (-\infty , -1] \cup [1 , +\infty) \)
   • period : \( 2\pi \)
   • \( \csc(x) \) has no zeros.
   • vertical asymptotes at \( x = k\pi \) where \( k \) is an integer.

TUTORIAL (2) - Relationship Between Basic Trigonometric Functions

1. \( \sin(x) = \cos(x-\pi/2) \) , \( \cos(x) = \sin(x+\pi/2) \)
2. Since \( \csc(x) = \dfrac{1}{\sin(x)} \) , the values of \( x \) that make \( \sin(x) = 0 \) will create a division by zeros for csc(x) meaning that there is a vertical asymptotes at the same \( x \) values.
3. Since \( \sec(x) = \dfrac{1}{\cos(x)} \), the values that make \( \cos(x) = 0 \) will create a division by zeros for sec(x) meaning that there is a vertical asymptotes at the same \( x \) values.
4. \[ \sin(x) = \cos(x-\pi/2) \] \[ \sec(x) = \dfrac{1}{\cos(x)} \] \[ \csc(x) = \dfrac{1}{\cos(x-\pi/2)} \] \[ \tan(x) = \dfrac{\cos(x-\pi/2)}{\cos(x)} \]\ \[ \cot(x) = \dfrac{\cos(x)}{\cos(x-\pi/2)} \]
5. \[ \cos(x) = \sin(x+\pi/2) \] \[ \sec(x) = \dfrac{1}{\sin(x+\pi/2)} \] \[ \csc(x) = \dfrac{1}{\sin(x)} \] \[ \tan(x) = \dfrac{\sin(x)}{\sin(x+\pi/2)} \] \[ \cot(x) = \dfrac{\sin(x+\pi/2)}{\sin(x)} \]

More references and links related to trigonometric functions and their properties.

• Graph Sine Function
• Sine Function
• Cosine Function
• Tangent Function
• Secant Function
• Unit Circle in Trigonometry and sin(x), cos(x) and tan(x)
• Solve Trigonometric Equations
• Graphs of Trigonometric Functions - Self Test