Trigonometry - sin, cos, tan, cot
Take an x-axis and an y-axis (orthonormal) and let O be the origin. A circle centered in O and with radius = 1 is known as trigonometric circle or unit circle.
- the x-coordinate of P is called the cosine of t. We write cos(t);
- the y-coordinate of P is called the sine of t. We write sin(t);
- the number sin(t)/cos(t) is called the tangent of t. We write tan(t);
- the number cos(t)/sin(t) is called the cotangent of t. We write cot(t).
The sine function
sin : R -> R
All trygonometric functions are periodic. The period of sin is 2π.
The range of the function is [-1,1].
The cosine function
cos : R -> R
The period of sin is 2π.
The range of the function is [-1,1].
The tangent function
tan : R -> R
The range of the function is R.
Now, the period is π and the function is undefined at x = (π/2) + kπ, k=0,1,2,...
The cotangent function
cot : R -> R
The range of the function isR.
The period is π and that the function is undefined at x = kπ, k=0,1,2,...
The values of sin, cos, tan, cot at the angles of 0°, 30°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°
The easiest way to remember the basic values of sin and cos
at the angles of 0°, 30°, 60°, 90°:
sin([0, 30, 45, 60, 90]) = cos([90, 60, 45, 30, 0]) = sqrt([0, 1, 2, 3, 4]/4)
Trigonometric identities
With t radians corresponds exactly one point P(cos(t),sin(t)) on the unit circle. The square of the distance [OP] = 1. Calculating this distance with the coordinates of P we have for each t:
If t + t' = 180° then:
- sin(t) = sin(t')
- cos(t) = -cos(t')
- tan(t) = -tan(t')
- cot(t) = -cot(t')
If t + t' = 90° then:
- sin(t) = cos(t')
- cos(t) = sin(t')
- tan(t) = cot(t')
- cot(t) = tan(t')
Trigonometric formulas
cos(u + v) = cos(u - (-v)) = cos(u).cos(-v) + sin(u).sin(-v)
| tan(u + v) = |
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= |
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| tan(u + v) = |
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cos(2u) = cos2(u) - sin2(u)
| tan(2u) = |
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| cos(2u) = |
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| sin(2u) = |
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1 - cos(2u) = 2 sin2(u)
Addition and multiplication of sin and cos
Trigonometric functions plotter
Trigonometry in the math forum
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