trigonometry

150 GEOMETRY AND TRIGONOMETRY
(vii) In the first quadrant of Fig. 15.1 all the curves
have positive values; in the second only sine is
positive; in the third only tangent is positive;
in the fourth only cosine is positive (exactly as
summarized in Fig. 15.4).
A knowledge of angles of any magnitude is needed
when finding, for example, all the angles between
0 ◦ and 360 ◦ whose sine is, say, 0.3261. If 0.3261
is entered into a calculator and then the inverse
sine key pressed (or sin −1 key) the answer 19.03 ◦
appears. However there is a second angle between
0 ◦ and 360 ◦ which the calculator does not give.
Sine is also positive in the second quadrant
(either from CAST or from Fig. 15.1(a)). The
other angle is shown in Fig. 15.5 as angle θ
where θ = 180 ◦ − 19.03 ◦ = 160.97 ◦ . Thus 19.03 ◦
and 160.97 ◦ are the angles between 0 ◦ and 360 ◦
whose sine is 0.3261 (check that sin 160.97 ◦ =
0.3261 on your calculator).
Figure 15.6
Problem 2. Determine all the angles between
0 ◦ and 360 ◦ whose tangent is 1.7629.
A tangent is positive in the first and third
quadrants (see Fig. 15.7(a)). From Fig. 15.7(b),
Figure 15.5
Be careful! Your calculator only gives you one
of these answers. The second answer needs to
be deduced from a knowledge of angles of any
magnitude, as shown in the following problems.
Problem 1. Determine all the angles between
0 ◦ and 360 ◦ whose sine is −0.4638.
The angles whose sine is −0.4638 occurs in the
third and fourth quadrants since sine is negative in
these quadrants (see Fig. 15.6(a)). From Fig. 15.6(b),
θ = sin −1 0.4638 = 27 ◦ 38 ′ .
Measured from 0 ◦ , the two angles between 0 ◦ and
360 ◦ whose sine is −0.4638 are 180 ◦ + 27 ◦ 38 ′ , i.e.
207 ◦ 38 ′ and 360 ◦ − 27 ◦ 38 ′ , i.e. 332 ◦ 22 ′ . (Note that
a calculator generally only gives one answer, i.e.
−27.632588 ◦ ).
Figure 15.7