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Wize High School Grade 12 Pre-Calculus Textbook > Trigonometric (Sinusoidal) Graphs

Graphs of Sinusoidal Functions

Solving Simple Trigonometric EquationsPrevious Section
Transformations of Sinusoidal FunctionsNext Section
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Graphs of Sinusoidal Functions

Sinusoidal functions are trigonometric functions such as sin⁡θ,cos⁡θ,tan⁡θ\sin{\theta},\cos{\theta},\tan{\theta}. These functions oscilliate and are also called periodic functions.

The Sine Function

y=sin⁡θy=\sin{\theta}

Minimums(3π2±2πn,−1),  n∈WMaximums(π2±2πn,1),  n∈WX-intercepts(0±πn,0),  n∈WAmplitude1Period2πDomain−∞<θ<∞Range−1≤y≤1\begin{array}{|c|c|}\hline\\ \textbf{Minimums}&\Bigg(\displaystyle\frac{3\pi}{2}\pm2\pi{}n,-1\Bigg),~~n\in\mathbb{W}\\\\\hline\\ \textbf{Maximums}&\Bigg(\displaystyle\frac{\pi}{2}\pm2\pi{}n,1\Bigg),~~n\in\mathbb{W}\\\\\hline\\ \textbf{X-intercepts}&(0\pm{}\pi{}n,0),~~n\in\mathbb{W}\\\\\hline\\ \textbf{Amplitude}&1\\\\\hline\\ \textbf{Period}&2\pi\\\\\hline\\ \textbf{Domain}&-\infin<\theta<\infin\\\\\hline\\ \textbf{Range}&-1\leq{}y\leq{}1\\\\\hline \end{array}
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The Cosine Function

y=cos⁡θy=\cos{\theta}

Minimums(π±2πn,−1),  n∈WMaximums(0±2nπ,1),  n∈WX-intercepts(π2±2πn,0),  n∈WAmplitude1Period2πDomain−∞<θ<∞Range−1≤y≤1\begin{array}{|c|c|}\hline\\ \textbf{Minimums}&(\pi\pm2\pi{}n,-1),~~n\in\mathbb{W}\\\\\hline\\ \textbf{Maximums}&(0\pm{}2n\pi,1),~~n\in\mathbb{W}\\\\\hline\\ \textbf{X-intercepts}&\Bigg(\displaystyle\frac{\pi}{2}\pm2\pi{}n,0\Bigg),~~n\in\mathbb{W}\\\\\hline\\ \textbf{Amplitude}&1\\\\\hline\\ \textbf{Period}&2\pi\\\\\hline\\ \textbf{Domain}&-\infin<\theta<\infin\\\\\hline\\ \textbf{Range}&-1\leq{}y\leq{}1\\\\\hline \end{array}

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The Tangent Function

y=tan⁡θy=\tan{\theta}

X-intercepts(0±πn,0),  n∈WVertical Asymptotesθ=π2±πn,  n∈WPeriodπDomain{θ∈R∣ θ≠π2±nπ,  n∈W}Range−∞<y<∞\begin{array}{|c|c|}\hline\\ \textbf{X-intercepts}&(0\pm{}\pi{}n,0),~~n\in\mathbb{W}\\\\\hline\\ \textbf{Vertical Asymptotes}&\theta=\displaystyle\frac{\pi}{2}\pm{}\pi{}n,~~n\in\mathbb{W}\\\\\hline\\ \textbf{Period}&\pi\\\\\hline\\ \textbf{Domain}&\{\theta\in\mathbb{R}|~\theta\neq{}\frac{\pi}{2}\pm{}n\pi,~~n\in\mathbb{W}\}\\\\\hline\\ \textbf{Range}&-\infin<y<\infin\\\\\hline \end{array}