Trigonometric Graphs

NZ Level 7 (NZC) Level 2 (NCEA)

Key features of sine and cosine curves

Lesson

When looking at the unit circle, that is a circle with the origin at the centre and unit radius, the coordinates of any point on that circle can be described using trigonometry. Specifically a point on the circle at an angle of $\theta$`θ` anticlockwise from the $x$`x`-axis has coordinates $\left(\cos\theta,\sin\theta\right)$(`c``o``s``θ`,`s``i``n``θ`).

A point on the unit circle |

As we move through different values of $\theta$`θ` the value of $\cos\theta$`c``o``s``θ` and $\sin\theta$`s``i``n``θ` move accordingly between $-1$−1 and $1$1. If we plot the values of $\cos\theta$`c``o``s``θ` and $\sin\theta$`s``i``n``θ` according to different values of $\theta$`θ` on the unit circle, we get the following graphs:

$y=\cos\theta$y=cosθ |

$y=\sin\theta$y=sinθ |

Consequently, the graphs of $y=\cos\theta$`y`=`c``o``s``θ` and $y=\sin\theta$`y`=`s``i``n``θ` have many properties. Each graph demonstrates repetition. We call the graphs of $y=\cos\theta$`y`=`c``o``s``θ` and $y=\sin\theta$`y`=`s``i``n``θ` cyclical and define a cycle as any section of the graph that can be translated to complete the rest of the graph. We also define the period as the length of one cycle. For both graphs, the period is $360^\circ$360°.

An example of a cycle |

Because of the oscillating behaviour, both graphs have regions where the curve is increasing and decreasing. Remember that we say the graph of a particular curve is increasing if the $y$`y`-values increase as the $x$`x`-values increase. Similarly, we say the graph is decreasing if the $y$`y`-values decrease as the $x$`x`-values increase.

An example of where $y=\sin x$y=sinx is decreasing |

In addition, the height of each graph stays between $y=-1$`y`=−1 and $y=1$`y`=1 for all values of $\theta$`θ`, since each coordinate of a point on the unit circle can be at most $1$1 unit from the origin.

Consider the curve $y=\sin x$`y`=`s``i``n``x` drawn below and answer the following questions.

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What is the $y$

`y`-intercept? Give your answer as coordinates in the form $\left(a,b\right)$(`a`,`b`).What is the maximum $y$

`y`-value?What is the minimum $y$

`y`-value?

Consider the curve $y=\cos x$`y`=`c``o``s``x` drawn below and determine whether the following statements are true or false.

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The graph of $y=\cos x$

`y`=`c``o``s``x`is cyclic.True

AFalse

BTrue

AFalse

BAs $x$

`x`approaches infinity, the height of the graph approaches infinity.True

AFalse

BTrue

AFalse

BThe graph of $y=\cos x$

`y`=`c``o``s``x`is increasing between $x=90^\circ$`x`=90° and $x=180^\circ$`x`=180°.True

AFalse

BTrue

AFalse

B

Consider the curve $y=\cos x$`y`=`c``o``s``x` drawn below and answer the following questions.

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If one cycle of the graph of $y=\cos x$

`y`=`c``o``s``x`starts at $x=\left(-90\right)^\circ$`x`=(−90)°, when does the next cycle start?In which of the following regions is the graph of $y=\cos x$

`y`=`c``o``s``x`decreasing? Select all that apply.$\left(-180\right)^\circ

(−180)°< `x`<0°A$\left(-360\right)^\circ

(−360)°< `x`<(−180)°B$0^\circ

0°< `x`<180°C$180^\circ

180°< `x`<360°D$\left(-180\right)^\circ

(−180)°< `x`<0°A$\left(-360\right)^\circ

(−360)°< `x`<(−180)°B$0^\circ

0°< `x`<180°C$180^\circ

180°< `x`<360°DWhat are the $x$

`x`-values of the $x$`x`-intercepts in the region $0^\circ0°< `x`<360°? Give your answers on the same line separated by a comma.

Display the graphs of linear and non-linear functions and connect the structure of the functions with their graphs

Apply graphical methods in solving problems