082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

3.4 The sine, cosine and tangent functions 121
f (x)
f (x)
1
1
p
– –2
p
–p
p
–
2
2p
3p
x
–p
p
2p
x
–1
–1
Figure3.7
Graphof f(x) = sinx.
Figure3.8
Graphof f(x) = cosx.
f (x)
0 p 2p x
Figure3.9
Graphof f(x) = tanx.
By shifting the cosine function to the right by an amount π/2 the sine function is
obtained. Similarly, shifting the sine function to the left by π/2 results in the cosine
function. This interchangeability between the sine and cosine functions is reflected in
their being commonly referred to as sinusoidal functions. Notice also from the graphs
two importantpropertiesofsinx and cosx:
sinx = −sin(−x)
cosx =cos(−x)
For example,
sin π (
3 = −sin − π )
3
and cos π (
3 = cos − π )
3
Note that f(x) = sinx is 0 whenx = 0,±π,±2π,±3π,..., ±nπ,.... Also f(x) =
cosxis0whenx=± π 2 ,±3π 2 ,±5π 2 ,..., ±(2n+1)π 2 ,....
3.4.1 Inversetrigonometricfunctions
Allsixtrigonometricfunctionshaveaninversebutwewillonlyexaminethoseofsinx,
cosxandtanx.Theinversefunctionsofsinx,cosxandtanxaredenotedsin −1 x,cos −1 x
and tan −1 x. This notation can and does cause confusion. The ‘−1’ in sin −1 x is sometimesmistakenlyinterpretedasapower.Wewrite
(sinx) −1 todenote1/sinx.Valuesof
sin −1 x, cos −1 x and tan −1 x can be found using a scientific calculator. Ify = sinx, then
x = sin −1 y, as in
sinx = 0.7654 x = sin −1 (0.7654) = 0.8717