Period & Related Angles (SQA National 5 Maths): Revision Note

Period & related angles

What are the periods of the trigonometric functions?

• The periods of the basic trigonometric functions are discussed at Graphs of Basic Trigonometric Functions

• $\mathit{y}\mathbf{=}\mathbf{sin}\mathit{x}$ has a period of 360° (it repeats every 360°)

  • If $\sin x$ has some value for $x=a$

    • then it has that same value for $x=a+360°,a+720°,$ etc.

    • and also for $x=a-360°,a-720°,$ etc.

  • E.g. $\sin \left(30\right)=0.5$

    • so $\sin \left(390\right)$ is also equal to $0.5$

• $\mathit{y}\mathbf{=}\mathbf{cos}\mathit{x}$ also has a period of 360° (it repeats every 360°)

  • If $\cos x$ has some value for $x=a$

    • then it has that same value for $x=a+360°,a+720°,$ etc.

    • and also for $x=a-360°,a-720°,$ etc.

  • E.g. $\cos \left(60\right)=0.5$

    • so $\cos \left(420\right)$ is also equal to $0.5$

• $\mathit{y}\mathbf{=}\mathbf{tan}\mathit{x}$ has a period of 180° (it repeats every 180°)

  • If $\tan x$ has some value for $x=a$

    • then it has that same value for $x=a+180°,a+360°,$ etc.

    • and also for $x=a-180°,a-360°,$ etc.

  • E.g. $\tan \left(45\right)=1$

    • so $\tan \left(225\right)$ is also equal to $1$

What are related angles?

• The sin and cos functions are both symmetric

  • This symmetry allows you to find other values for x that give the same values of sin or cos

• These 'other values of $x$' are known as related angles

How do I use related angles with sin x?

• Sketch the sine graph for the given interval

  • Identify the first value on the graph

    • then use the symmetry of the graph to find additional values

  • E.g. if you know $\sin \left(30\right)=0.5$ and want to find other values of $x$for which $\sin x=0.5$, where $0°\le x\le 360°$

    • Sketch the graph $y=\sin x$ for $0°\le x\le 360°$

    • Draw on $\sin \left(30\right)=0.5$

    • By symmetry, another value of $x$ is $180°-30°=150°$

    • $\sin \left(150\right)$ is also equal to 0.5

How do I use related angles with cos x?

• Sketch the cosine graph for the given interval

  • Identify the first value on the graph

    • then use the symmetry of the graph to find additional values

  • E.g. if you know $\cos \left(60\right)=0.5$ and want to find other values of $x$for which $\cos x=0.5$, where $0°\le x\le 360°$

    • Sketch the graph $y=\cos x$ for $0°\le x\le 360°$

    • Draw on $\cos \left(60\right)=0.5$

    • By symmetry, another value of $x$ is $360°-60°=300°$

    • $\cos \left(300\right)$ is also equal to 0.5