Arccos(x), cos-1(x), inverse cosine function.
Arccos definition
The arccosine of x is defined as the inverse cosine function of x when -1≤x≤1.
When the cosine of y is equal to x:
cos y = x
Then the arccosine of x is equal to the inverse cosine function of x, which is equal to y:
arccos x = cos-1 x = y
Here cos-1 x means the inverse cosine and does not mean cosine to the power of -1.
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For example:
arccos 1 = cos-1 1 = 0 rad = 0°
Graph of arccos
Arccos rules
Rule name | Rule |
Cosine of arccosine | cos( arccos x ) = x |
Arccosine of cosine | arccos( cos x ) = x + 2kπ, when k∈ℤ (k is integer) |
Arccos of negative argument | arccos(-x) = π - arccos x = 180° - arccos x |
Complementary angles | arccos x = π/2 - arcsin x = 90° - arcsin x |
Arccos sum | arccos(α) + arccos(β) = arccos( αβ - √(1-α2)(1-β2) ) |
Arccos difference | arccos(α) - arccos(β) = arccos( αβ + √(1-α2)(1-β2) ) |
Arccos of sin of x | arccos( sin x ) = -x - (2k+0.5)π |
Sine of arccosine | cos (arcsin x) = sin (arccos x) = √1 - x2 |
Tangent of arccosine | |
Derivative of arccosine | |
Indefinite integral of arccosine | ∫ arccos x dx = x arccos x - √1-x2 + C |
Arccos table
x | arccos(x)(rad) | arccos(x)(°) |
-1 | π | 180° |
-√3/2 | 5π/6 | 150° |
-√2/2 | 3π/4 | 135° |
-1/2 | 2π/3 | 120° |
0 | π/2 | 90° |
1/2 | π/3 | 60° |
√2/2 | π/4 | 45° |
√3/2 | π/6 | 30° |
1 | 0 | 0° |
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