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The trigonometric ratios are the ratios between the two sides of a right-angles triangle with respect to an angle and hence they are real numbers.
The angle θ taken into consideration may be acute, obtuse or right angle.
1) sin2 θ + cos2 θ = 1
2) sec2 θ -tan2 θ = 1
3) cosec2 θ - cot2 θ = 1
4) sin θ cosec θ = tan θ cot θ = sec θ cos θ = 1
5) sin2 θ + cos2 θ = 1, so each of them is numerically less than 1.
6) |sin θ| ≤ 1 and |cos θ| ≤ 1
7) -1 <cosec θ<1 and -1 <sec θ<1
8) tan θ and cot θ may take any value.
Range f = {f(x) ∈ Y: x ∈ X} ⊆ Y.
The domain and range of various trigonometry ratios are given below:
Trigonometric Function
Domain
Range
sin x
R
-1 ≤ sin x ≤1
cos x
-1 ≤ cos x ≤1
tan x
R – {(2n + 1)π/2, n ∈ I}
cosec x
R – {nπ, n ∈ I}
R – {x: -1 < x <1}
sec x
cot x
Angle (x)
0°
0
1
undefined
90° = π/2
180° = π
-1
270° =3π/2
360° = 2π
Angles
30°
45°
60°
90°
sin
1/2
1/√2
√3/2
cos
tan
√3
cosec
2
√2
2/√3
sec
cot
1/√3
First Quadrant
1. sine- increases from 0 to 1
2. cosine- decreases from 1 to 0
3. tangent- increases from 0 to ∞
4. cotangent- decreases from ∞ to 0
5. secant- increases from 1 to ∞
6. cosecant- decreases from ∞ to 1
Second Quadrant
1. sine- decreases from 1 to 0
2. cosine- decreases from 0 to -1
3. tangent- increases from -∞ to 0
4. cotangent- decreases from 0 to -∞
5. secant- increases from -∞ to -1
6. cosecant- increases from 1 to ∞
Third Quadrant
1. sine- decreases from 0 to -1
2. cosine- increases from -1 to 0
5. secant- decreases from -1 to -∞
6. cosecant- increases from -∞ to -1
Fourth Quadrant
1. sine- increases from -1 to 0
2. cosine- increases from 0 to 1
5. secant- decreases from ∞ to 1
6. cosecant- decreases from -1 to ∞
Sin and cos both have 2π as their period.
tan function has π as the period.
The reciprocal functions have the same period as that of the original functions.
If the period of f(x) is T then that of kf(ax+b) is T/mod (a), hence period is affected by coefficient of x only.
If f(x) has its period T and g(x) has its period M, then (af(x) + bg(x)) has its period < L.C.M. (T, M). Moreover if f(x) and g(x) are basic trigonometric functions then period of [af (x) + bg(x)] = L.C.M. (T, M).
If a constant is added, subtracted, multiplied or divided in a periodic function, its period is unaffected.
1) sin x
2) cos x
3) tan x
4) cot x
5) sec x
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Solved Examples on Trigonometric Functions...