Revision of The Sine Function

Functions of the form \(y = \sin \theta\) for \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\)

• Period of one complete wave is \(\text{360}\)\(\text{°}\).

• Amplitude is the maximum height of the wave above and below the \(x\)-axis and is always positive. Amplitude = \(\text{1}\).

• Domain: \([\text{0}\text{°};\text{360}\text{°}]\)

  For \(y = \sin \theta\), the domain is \(\{ \theta: \theta \in \mathbb{R} \}\), however in this case, the domain has been restricted to the interval \(\text{0}\text{°} \leq \theta \leq \text{360}\text{°}\).

• Range: \([-1;1]\)

• \(x\)-intercepts: \((\text{0}\text{°};0)\), \((\text{180}\text{°};0)\), \((\text{360}\text{°};0)\)

• \(y\)-intercept: \((\text{0}\text{°};0)\)

• Maximum turning point: \((\text{90}\text{°};1)\)

• Minimum turning point: \((\text{270}\text{°};-1)\)

Functions of the form \(y = a \sin \theta + q\)

The effects of \(a\) and \(q\) on \(f(\theta) = a \sin \theta + q\):

• The effect of \(q\) on vertical shift

  • For \(q>0\), \(f(\theta)\) is shifted vertically upwards by \(q\) units.

  • For \(q<0\), \(f(\theta)\) is shifted vertically downwards by \(q\) units.

• The effect of \(a\) on shape

  • For \(a>1\), the amplitude of \(f(\theta)\) increases.

  • For \(0<a<1\), the amplitude of \(f(\theta)\) decreases.

  • For \(a<0\), there is a reflection about the \(x\)-axis.

  • For \(-1 < a < 0\), there is a reflection about the \(x\)-axis and the amplitude decreases.

  • For \(a < -1\), there is a reflection about the \(x\)-axis and the amplitude increases.