# 10.3 Graphs of `y` = `a` sin( `bx` + `c`) and `y` = `a` cos(`bx` + `c`)

**Phase Angle • Displacement • Graphs of $y\hspace{0.17em}=\hspace{0.17em}a\text{}\mathrm{sin}\text{}(bx\hspace{0.17em}+\hspace{0.17em}c)$ and $y\hspace{0.17em}=\hspace{0.17em}a\text{}\mathrm{cos}\text{}(bx\hspace{0.17em}+\hspace{0.17em}c)$ • Cycle**

In the function $y\hspace{0.17em}=\hspace{0.17em}a\text{}\mathrm{sin}\text{}(bx\hspace{0.17em}+\hspace{0.17em}c)\hspace{0.17em},\hspace{0.17em}$ `c` represents the **phase angle**. It is another very important quantity in graphing the sine and cosine functions. Its meaning is illustrated in the following example.

# EXAMPLE 1 Sketch function with phase angle

Sketch the graph of $y\hspace{0.17em}=\hspace{0.17em}\mathrm{sin}(2x\hspace{0.17em}+\hspace{0.17em}{\displaystyle \frac{\pi}{4}})\hspace{0.17em}.\hspace{0.17em}$

Here, $c\hspace{0.17em}=\hspace{0.17em}\pi \hspace{0.17em}/\hspace{0.17em}4\hspace{0.17em}.\hspace{0.17em}$ Therefore, in order to obtain values for the table, we assume a value for `x`, multiply it by 2, add $\pi \hspace{0.17em}/\hspace{0.17em}4$ to this value, and then find the sine of the result. The values shown are those for which $2x\hspace{0.17em}+\hspace{0.17em}\pi \hspace{0.17em}/\hspace{0.17em}4\hspace{0.17em}=\hspace{0.17em}0\hspace{0.17em},\hspace{0.17em}\text{}\pi \hspace{0.17em}/\hspace{0.17em}4\hspace{0.17em},\hspace{0.17em}\text{}\pi \hspace{0.17em},\hspace{0.17em}$ 2, $3\pi \hspace{0.17em}/\hspace{0.17em}4\hspace{0.17em},\hspace{0.17em}\text{}\pi \hspace{0.17em},\hspace{0.17em}$ and so on, which are the important values for $y\hspace{0.17em}=\hspace{0.17em}\mathrm{sin\; ...}$

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