The sine, cosine and tangents of common angles:
These occur frequently and need to be remembered.
Quadrants and the 'cast' rule
On a set of axes, angles are measured anti-clockwise from the positive x-axis. So 30º would be drawn as follows:
The angles which lie between 0º and 90º are said to lie in the first quadrant. The angles between 90º and 180º are in the second quadrant, angles between 180º and 270º are in the third quadrant and angles between 270º and 360º are in the fourth quadrant:
In the first quadrant, the values for sin, cos and tan are positive.
In the second quadrant, the values for sin are positive only.
In the third quadrant, the values for tan are positive only.
In the fourth quadrant, the values for cos are positive only.
This can be summed up as follows:
In the fourth quadrant, Cos is positive, in the first, All are positive, in the second, Sin is positive and in the fourth quadrant, Tan is positive. This is easy to remember, since it spells 'cast'.
The sines, cosines and tangents of some angles are equal to the sines, cosines and tangents of other angles. For example, cos(-30º) = cos(30º) and cos(30º) = cos(390º) . In the following diagrams, the sines, cosines and tangents of each of the shaded angles have the same magnitude (ø is the same angle in each diagram):
For example, if ø is 30º,
sin30º = 0.5
sin150º = 0.5
sin210º = -0.5
sin330º = -0.5
These angles are 'related angles' and their cosines and tangents will be related in a similar way. Note that the signs of the sines (/cosines/tangents) are found using the 'cast' rule.
Arcsin, arccos, arctan
Arcsin is another way of writing the inverse of sin, arccos means the inverse of cos and arctan means the inverse of tan. For example, arcsin(0.5) = 30º . However, although this is true, we also know that sin(150º) = 0.5 (using the idea of related angles and the 'cast rule'). If we continue moving round the 'unit circle' (the circle with radius 1 that we have been drawing angles on above), then we find that sin(390º) is also 0.5 .
So we can write arcsin(0.5) = 30º, 150º, 390º, ...
Solve the equation sinø = 0.6428, for 0 < ø < 360º
therefore ø = arcsin(0.6428)
= 40º, 140º, 400º, ...
but the question asks for solutions between 0 and 360º, so the answer is 40º and 140º .
Graphing sinø, cosø and tanø
The following are graphs of sinø, cosø and tanø:
Points to note:
The graphs of sinø and cosø are periodic, with period of 360º (in other words the graphs repeat themselves every 360º .
The graph of cosø is the same as the graph of sinø, although it is shifted 90º to the right/ left. For this reason sinø = cos(90 - ø) or cosø = sin(90 - ø).
The graph of tanø has asymptotes. An asymptote is a line which the graph gets very close to, but does not touch. The red lines are asymptotes.
These graphs obey the usual laws of graph transformations.