More about trigonometry - (13.1-13.3)

Understand the functions sine, cosine and tangent, and their graphs and properties, including maximum and minimum values and periodicity

  • Theory
    functions of sine..
  • Examples
    In a right-angled triangle, if \(\tan \theta = \frac{{60}}{{91}}\),
    find the values of \(\sin \theta \) and \(\cos \theta \).
  • Solutions
    By Pythagoras’ theorem,
    \(\begin{array}{c}AB = \sqrt {{{60}^2} + {{91}^2}} \\ = 109\end{array}\)
    \(\sin \theta = \underline{\underline {\frac{{60}}{{109}}}} \)
    \(\cos \theta = \underline{\underline {\frac{{91}}{{109}}}} \)
  • Graph
  • Theory
    sin graph, cos graph...
  • Examples
    The figure shows the graph of \(y = \sin 2x - \cos 2x{\rm{ }},\) where \(0^\circ \le x \le 180^\circ \).

    Solve the equation \(\sin 2x - \cos 2x = 0\) graphically,
    where \(0^\circ \le x \le 180^\circ \). (Give your answers correct to the nearest \(\ 4^\circ \) if necessary.)
  • Solutions
    From the graph, when \(y = 0\),
    \(x = 24^\circ \) or \(112^\circ \) (corr. to the nearest \(\ 4^\circ \) )
  • Theory
    properities of sin..
  • Examples
    Express the following in terms of trigonometric ratios of acute angles.
    a) \(\sin 217^\circ \)
    b) \(\cos 343^\circ \)
    c) \(\tan ( - {\rm{ }}122^\circ )\)
  • Solutions
    \(\begin{array}{c} a)\qquad \sin 217^\circ = \sin (180^\circ + 37^\circ )\\ = {\underline{\underline { - {\rm{ }}\sin 37^\circ }} ^{}}\end{array}\)

    \(\begin{array}{c} b)\qquad \cos 343^\circ = \cos (360^\circ - 17^\circ )\\ = {\underline{\underline {\cos 17^\circ }} ^{}}\end{array}\)

    \(\begin{array}{c} c)\qquad \tan ( - {\rm{ }}122^\circ ) = \tan 238^\circ \\ = \tan {(180^\circ + 58^\circ )^{}}\\ = {\underline{\underline {\tan 58^\circ }} ^{}}\end{array}\)
  • Theory
    maximum and minimum...
  • Examples
    Find the maximum and minimum values of each of the following functions.
    \(y = 2\sin x^\circ + 1\)

  • Solutions
    For any real number x,
    \(\begin{array}{c} - {\rm{ }}1 \le \sin x^\circ \le 1\\ - {\rm{ }}2 \le 2\sin x^\circ \le 2\\ - {\rm{ }}1 \le 2\sin x^\circ + 1 \le 3\end{array}\)
    Maximum value of the function\( = \underline{\underline 3} \)
    Minimum value of the function\( = \underline{\underline { - {\rm{ }}1}} \)

  • Theory
    periodicity
  • Examples
    Find the period of each of the following functions.
    \(y = \sin (x - 50)^\circ \)
  • Solutions
    \(\begin{array}{1} y= \sin (x - 50)^\circ \\ = \sin (x - 50 + 360){^\circ {}}\\ = \sin [(x + 360) - 50]{^\circ {}}\end{array}\)
    \(\therefore\) The period of the function is 360.

Solve the trigonometric equations a sin θ = b , a cos θ = b , a tan θ = b (solutions in the interval from 0° to 360° ) and other trigonometric equations (solutions in the interval from 0° to 360° )

  • Theory
  • Examples
    Solve the following equations, where \(0^\circ \le x \le 360^\circ \). (Give your answers correct to 1 decimal place.)
    a) \(7\tan x = 6\)
    b) \(12\cos x = - {\rm{ }}5\)
  • Solutions
    \(\begin{array}{c} a)\qquad 7\tan x = 6\\\tan x = \frac{6}{7}\end{array}\)
    \(x = 40.6^\circ \) or \(180^\circ + 40.6^\circ \) (corr. to 1 d.p.)
    \( = 40.6^\circ \) or \(220.6^\circ \)

    \(\begin{array}{c} b)\qquad 12\cos x = - 5\\\cos x = - {\rm{ }}\frac{5}{{12}}\end{array}\)
    \(x = 180^\circ - 65.4^\circ \) or \(180^\circ + 65.4^\circ \) (corr. to 1 d.p.)
    \( = 114.6^\circ \) or \(245.4^\circ \)

Understand the formula ½ ab sin C for areas of triangles

  • Theory
  • Examples
    In the figure, if the area of △ABC is \(25{\rm{ c}}{{\rm{m}}^2}\), find the value of x.
  • Solutions

    \(\begin{array}{c} area= 25\\ \frac{1}{2}(10)(x)\sin 120^\circ = 25\\(\frac{{5\sqrt 3 }}{2})x = 25\\x = \frac{{10}}{{\sqrt 3 }}\\ x= {\frac{{10\sqrt 3 }}{3}^{}}\end{array}\)