Math 140

Class 11

Properties of sine and cosine

• Domain: \(\mathbb R\) (the whole number circle)
• Range: \(\cos\): \([-1,1]\) on the \(x\)-axis, \(\sin\): \([-1,1]\) on the \(y\)-axis
• Symmetry: \(\cos\) is even, \(\sin\) is odd
• Both are \(2\pi\)-periodic: \(\cos(t + 2\pi) = \cos t\) and \(\sin(t + 2\pi) = \sin t\)
• \(\cos(t + \pi) = -\cos t\) and \(\sin(t + \pi) = -\sin t\)
• Intercepts (zeros): \(\cos t = 0\) at \(t = (2k + 1)\frac{\pi}{2}\), \(\sin t = 0\) at \(t = k\pi\)
• One-to-one:
  • \(\cos t\) is one-to-one on any interval \([k\pi, (k+1)\pi]\) (upper or lower semicircle).
  • \(\sin t\) is one-to-one on any interval \(\left[(2k-1)\frac{\pi}{2}, (2k+1)\frac{\pi}{2}\right]\) (right or left semicircle).
• Pythagorean Identity: \(\cos^2 t + \sin^2 t = 1\) for any \(t\).
• Co-function Identities: \(\cos(t) = \sin\left(\frac{\pi}{2} - t\right)\) and \(\sin(t) = \cos\left(\frac{\pi}{2} - t\right)\)

Properties of tangent and cotangent

• Domain:
  • \(\tan\): \(\displaystyle \bigcup_{k = -\infty}^\infty \left(\left(2k-1\right)\frac{\pi}{2}, \left(2k+1\right)\frac{\pi}{2}\right)\), left and right semicircles.
  • \(\cot\): \(\displaystyle \bigcup_{k = -\infty}^\infty \left(k\pi, \left(k+1\right)\pi\right)\), upper and lower semicircles.
• Vertical Asymptotes:
  • \(\tan\): \(x = (2k+1)\frac{\pi}{2}\) for any integer k.
  • \(\cot\): \(x = k\pi\) for any integer k.
• Range: \(\mathbb{R}\)
• Symmetry: both are odd
• Both are \(\pi\)-periodic: \(\tan(t + \pi) = \tan t\) and \(\cot(t + \pi) = \cot t\)

Properties of tangent and cotangent

• Intercepts (zeros): \(\cot t = 0\) at \(t = (2k + 1)\frac{\pi}{2}\), \(\tan t = 0\) at \(t = k\pi\)
• One-to-one:
  • \(\tan t\) is one-to-one (increasing) on any interval \(\left[(2k+1)\frac{\pi}{2}, (2k+3)\frac{\pi}{2}\right]\) (right or left semicircle).
  • \(\cot t\) is one-to-one (decreasing) on any interval \([k\pi, (k+1)\pi]\) (upper or lower semicircle).
• Co-function Identities: \(\cot(t) = \tan\left(\frac{\pi}{2} - t\right)\) and \(\tan(t) = \cot\left(\frac{\pi}{2} - t\right)\)

Properties of secant and cosecant

• Domain:
  • \(\sec\): \(\displaystyle \bigcup_{k = -\infty}^\infty \left(\left(2k-1\right)\frac{\pi}{2}, \left(2k+1\right)\frac{\pi}{2}\right)\), left and right semicircles.
  • \(\csc\): \(\displaystyle \bigcup_{k = -\infty}^\infty \left(k\pi, \left(k+1\right)\pi\right)\), upper and lower semicircles.
• Vertical Asymptotes:
  • \(\sec\): \(x = (2k+1)\frac{\pi}{2}\) for any integer k.
  • \(\csc\): \(x = k\pi\) for any integer k.
• Range: \((-\infty, -1] \cup [1, \infty)\)
• Symmetry: \(\sec\) is even, \(\csc\) is odd
• Both are \(2\pi\)-periodic: \(\sec(t + 2\pi) = \sec t\) and \(\csc(t + 2\pi) = \csc t\)
• \(\sec(t + \pi) = -\sec t\) and \(\csc(t + \pi) = -\csc t\)

Graph of cosine

Graph of sine

Transformations of Cosine and Sine

\(f(t) = a\cos\left(\omega t + \phi\right) + b\)

\(g(t) = a\sin\left(\omega t + \phi\right) + b\)

with \(\omega > 0\).

1. Vertical scaling by the factor \(a\) (with vertical flip is \(a < 0\)).
2. Horizontal shift by \(-\phi\) (left if \(c > 0\), right if \(c < 0\)).
3. Horizontal scaling by the factor \(1/\omega\).
4. Vertical shift by \(b\).

\(f(t) = a\cos\left(\omega t + \phi\right) + b\)

\(g(t) = a\sin\left(\omega t + \phi\right) + b\)

\(f(t) = a\cos\left(\omega t + \phi\right) + b\) with negative \(a\)

\(g(t) = a\sin\left(\omega t + \phi\right) + b\) with negative \(a\)

Transformations of Cosine and Sine

\(f(t) = a\cos\left(\omega t + \phi\right) + b\), \(\qquad g(t) = a\sin\left(\omega t + \phi\right) + b\)

• Amplitude \({}= \left\lvert a\right\rvert\)
• Period \({}= \frac{2\pi}{\omega}\)
• Phase shift \({}= -\frac{\phi}{\omega}\)
• Phase \({} = \phi\)
• Vertical shift \({}=b\)
  • baseline: \(x = b\) for cosine
  • baseline: \(y = b\) for sine
• Angular frequency \({} = \omega\)
• Frequency \({} = \frac{\omega}{2\pi} = \frac{1}{\text{period}}\)

Example