P1^
7
Trigonometrical(^) functions
(^) for
(^) angles
(^) of
(^) any
(^) size
First look at the right-angled triangle in figure 7.10 which has hypotenuse of
unit length.
This gives rise to the definitions:
sin;θ== cos;θ== tan.θ=
y
y
x
x
y
1 1 x
Now think of the angle θ being situated at the origin, as in figure 7.11, and allow
θ to take any value. The vertex marked P has co-ordinates (x, y) and can now be
anywhere on the unit circle.
You can now see that the definitions above can be applied to any angle θ, whether
it is positive or negative, and whether it is less than or greater than 90°
sin,θ=y cos,θ=x tan.θ=
y
x
For some angles, x or y (or both) will take a negative value, so the sign of sin θ,
cos θ and tan θ will vary accordingly.
ACTIvITy 7.1 Draw x and y axes. For each of the four quadrants formed, work out the sign of
sin θ, cos θ and tan θ, from the definitions above.
Identities involving sin θ, cos θ and tan θ
Since tan θ = (^) xy and y = sin θ and x = cos θ it follows that
tan θ = (^) cosinsθθ.
It would be more accurate here to use the identity sign, ≡, since the relationship
is true for all values of θ
tan θ ≡ (^) cosinsθθ.
An identity is different from an equation since an equation is only true for certain
values of the variable, called the solution of the equation. For example, tan θ = 1 is
O x
P
(^1) y
θ
Figure 7.10 O x x
P(x, y)
1
y
y
θ
Figure 7.11