### 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

• functions of sine..
• In a right-angled triangle, if $$\tan \theta = \frac{{60}}{{91}}$$,
find the values of $$\sin \theta$$ and $$\cos \theta$$.
• By Pythagoras’ theorem,
$$\begin{array}{c}AB = \sqrt {{{60}^2} + {{91}^2}} \\ = 109\end{array}$$
$$\sin \theta = \underline{\underline {\frac{{60}}{{109}}}}$$
$$\cos \theta = \underline{\underline {\frac{{91}}{{109}}}}$$
• sin graph, cos graph...
• 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.)
• From the graph, when $$y = 0$$,
$$x = 24^\circ$$ or $$112^\circ$$ (corr. to the nearest $$\ 4^\circ$$ )
• properities of sin..
• Express the following in terms of trigonometric ratios of acute angles.
a) $$\sin 217^\circ$$
b) $$\cos 343^\circ$$
c) $$\tan ( - {\rm{ }}122^\circ )$$
• $$\begin{array}{c} a)\qquad \sin 217^\circ = \sin (180^\circ + 37^\circ )\\ = {\underline{\underline { - {\rm{ }}\sin 37^\circ }} ^{}}\end{array}$$

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

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

• 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$$= \underline{\underline { - {\rm{ }}1}}$$

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

• 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) $$12\cos x = - {\rm{ }}5$$
• $$\begin{array}{c} a)\qquad 7\tan x = 6\\\tan x = \frac{6}{7}\end{array}$$
$$x = 40.6^\circ$$ 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

• In the figure, if the area of △ABC is $$25{\rm{ c}}{{\rm{m}}^2}$$, find the value of x.
• $$\begin{array}{c} area= 25\\ \frac{1}{2}(10)(x)\sin 120^\circ = 25\\(\frac{{5\sqrt 3 }}{2})x = 25\\x = \frac{{10}}{{\sqrt 3 }}\\ x= {\frac{{10\sqrt 3 }}{3}^{}}\end{array}$$