Graphs of Basic Trigonometric Functions

Click on the radio button of \( \sin(x) \) and use the graph to determine the range of \( \sin(x) \). What is the domain of \( \sin(x) \)? What are the zeros of \( \sin(x) \)?

Answer & Properties:

• Domain: \( (-\infty, +\infty) \)
• Range: \( [-1, 1] \)
• Period: \( 2\pi \)
• Zeros: at \( x = k\pi \), where \( k \) is an integer (\( k = 0, \pm 1, \pm 2, \dots \))

Click on the radio button of \( \cos(x) \) and use the graph to determine the range of \( \cos(x) \). What is the domain of \( \cos(x) \)? What are the zeros of \( \cos(x) \)?

Answer & Properties:

• Domain: \( (-\infty, +\infty) \)
• Range: \( [-1, 1] \)
• Period: \( 2\pi \)
• Zeros: at \( x = \dfrac{\pi}{2} + k\pi \), where \( k \) is an integer (\( k = 0, \pm 1, \pm 2, \dots \))

Click on the radio button of \( \tan(x) \). The red broken lines are the vertical asymptotes for the graph of \( \tan(x) \). Use the identity \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \] to find the domain of \( \tan(x) \). Find the vertical asymptotes and zeros of \( \tan(x) \), and use the graph to determine its range.

Answer & Properties:

• Domain: All real numbers except \( \dfrac{\pi}{2} + k\pi \) where \( k \) is an integer.
• Range: \( (-\infty, +\infty) \)
• Period: \( \pi \)
• Zeros: at \( x = k\pi \) (same zeros as \( \sin(x) \) because \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)).
• Vertical Asymptotes: at \( x = \dfrac{\pi}{2} + k\pi \) where \( k \) is an integer.

Click on the radio button of \( \cot(x) \). Use the identity \[ \cot(x) = \frac{\cos(x)}{\sin(x)} \] to find the domain, zeros, vertical asymptotes, and range of \( \cot(x) \).

Answer & Properties:

• Domain: All real numbers except \( k\pi \) where \( k \) is an integer.
• Range: \( (-\infty, +\infty) \)
• Period: \( \pi \)
• Zeros: at \( x = \dfrac{\pi}{2} + k\pi \) (same zeros as \( \cos(x) \) because \( \cot(x) = \frac{\cos(x)}{\sin(x)} \)).
• Vertical Asymptotes: at \( x = k\pi \) where \( k \) is an integer.

Click on the radio button of \( \sec(x) \). Use the identity \[ \sec(x) = \frac{1}{\cos(x)} \] to find the domain, range, period, zeros, and vertical asymptotes of \( \sec(x) \).

Answer & Properties:

• Domain: All real numbers except \( \dfrac{\pi}{2} + k\pi \) where \( k \) is an integer.
• Range: \( (-\infty, -1] \cup [1, +\infty) \)
• Period: \( 2\pi \)
• Zeros: \( \sec(x) \) has no zeros.
• Vertical Asymptotes: at \( x = \dfrac{\pi}{2} + k\pi \) where \( k \) is an integer.

Click on the radio button of \( \csc(x) \). Use the identity \[ \csc(x) = \frac{1}{\sin(x)} \] to find the domain, range, period, zeros, and vertical asymptotes of \( \csc(x) \).

Answer & Properties:

• Domain: All real numbers except \( k\pi \) where \( k \) is an integer.
• Range: \( (-\infty, -1] \cup [1, +\infty) \)
• Period: \( 2\pi \)
• Zeros: \( \csc(x) \) has no zeros.
• Vertical Asymptotes: at \( x = k\pi \) where \( k \) is an integer.

1. Phase Shift Relationships: \[ \sin(x) = \cos\left(x - \frac{\pi}{2}\right) \quad \text{and} \quad \cos(x) = \sin\left(x + \frac{\pi}{2}\right) \]
2. Sine & Cosecant Connection: Since \( \csc(x) = \dfrac{1}{\sin(x)} \), the values of \( x \) that make \( \sin(x) = 0 \) cause division by zero, creating vertical asymptotes at those exact locations.
3. Cosine & Secant Connection: Since \( \sec(x) = \dfrac{1}{\cos(x)} \), the values of \( x \) that make \( \cos(x) = 0 \) cause division by zero, resulting in vertical asymptotes at those same coordinates.
4. All Trigonometric Functions Expressed in Terms of Cosine Only: \[ \sin(x) = \cos\left(x - \frac{\pi}{2}\right) \] \[ \sec(x) = \frac{1}{\cos(x)} \] \[ \csc(x) = \frac{1}{\cos\left(x - \frac{\pi}{2}\right)} \] \[ \tan(x) = \frac{\cos\left(x - \frac{\pi}{2}\right)}{\cos(x)} \] \[ \cot(x) = \frac{\cos(x)}{\cos\left(x - \frac{\pi}{2}\right)} \]
5. All Trigonometric Functions Expressed in Terms of Sine Only: \[ \cos(x) = \sin\left(x + \frac{\pi}{2}\right) \] \[ \sec(x) = \frac{1}{\sin\left(x + \frac{\pi}{2}\right)} \] \[ \csc(x) = \frac{1}{\sin(x)} \] \[ \tan(x) = \frac{\sin(x)}{\sin\left(x + \frac{\pi}{2}\right)} \] \[ \cot(x) = \frac{\sin\left(x + \frac{\pi}{2}\right)}{\sin(x)} \]