11.1 Graphs, trigonometric identities, and solving trigonometric  (Page 6/12)

Complementary angles are positive acute angles that add up to ${90}^{\circ }$ . e.g. ${20}^{\circ }$ and ${70}^{\circ }$ are complementary angles.

Sine and cosine are known as co-functions . Two functions are called co-functions if $f\left(A\right)=g\left(B\right)$ whenever $A+B={90}^{\circ }$ (i.e. $A$ and $B$ are complementary angles). The other trig co-functions are secant and cosecant, and tangent and cotangent.

The function value of an angle is equal to the co-function of its complement (the co-co rule).

Thus for sine and cosine we have

Function values of $(\theta -{90}^{\circ })$

$sin(\theta -{90}^{\circ })=-cos\theta$ and $cos(\theta -{90}^{\circ })=sin\theta$ .

These results may be proved as follows:

and likewise for $cos(\theta -{90}^{\circ })=sin\theta$

Summary

The following summary may be made

Function values of $({270}^{\circ }\pm \theta )$

Angles in the third and fourth quadrants may be written as ${270}^{\circ }\pm \theta$ with $\theta$ an acute angle. Similar rules to the above apply. We get

Solving trigonometric equations

In Grade 10 and 11 we focussed on the solution of algebraic equations and excluded equations that dealt with trigonometric functions (i.e. $sin$ and $cos$ ). In this section, the solution of trigonometric equations will be discussed.

The methods described in previous Grades also apply here. In most cases, trigonometric identities will be used to simplify equations, before finding the final solution. The final solution can be found either graphically or using inverse trigonometric functions.

Graphical solution

As an example, to introduce the methods of solving trigonometric equations, consider

As explained in previous Grades,the solution of Equation [link] is obtained by examining the intersecting points of the graphs of:

Both graphs, for $-{720}^{\circ }<\theta <{720}^{\circ }$ , are shown in [link] and the intersection points of the graphs are shown by the dots.

In the domain for $\theta$ of $-{720}^{\circ }<\theta <{720}^{\circ }$ , there are 8 possible solutions for the equation $sin\theta =0,5$ . These are $\theta =[-{690}^{\circ },-{570}^{\circ },-{330}^{\circ },-{210}^{\circ },{30}^{\circ },{150}^{\circ },{390}^{\circ },{510}^{\circ }]$