Mathematics » Functions II » The Sine Function

Revision of The Sine Function

Revision of The Sine Function

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

Revision of The Sine Function

  • 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.

Revision of The Sine Function

Revision of The Sine Function

Do you want to suggest a correction or an addition to this content? Leave Contribution

[Attributions and Licenses]


This is a lesson from the tutorial, Functions II and you are encouraged to log in or register, so that you can track your progress.

Log In

Share Thoughts


Do NOT follow this link or you will be banned from the site!