# cacosf, cacos, cacosl

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< c‎ | numeric‎ | complex

Defined in header `<complex.h>` |  |
float complex cacosf( float complex z ); | (1) | (since C99)
double complex cacos( double complex z ); | (2) | (since C99)
long double complex cacosl( long double complex z ); | (3) | (since C99)
Defined in header `<tgmath.h>` |  |
#define acos( z ) | (4) | (since C99)

1-3) Computes the complex arc cosine of `z` with branch cuts outside the interval [−1,+1] along the real axis.
4) Type-generic macro: If `z` has type long double complex, `cacosl` is called. if `z` has type double complex, `cacos` is called, if `z` has type float complex, `cacosf` is called. If `z` is real or integer, then the macro invokes the corresponding real function (acosf, acos, acosl). If `z` is imaginary, then the macro invokes the corresponding complex number version.

### Parameters

z | - | complex argument

### Return value

If no errors occur, complex arc cosine of `z` is returned, in the range [0 ; ∞) along the real axis and in the range [−iπ ; iπ] along the imaginary axis.

### Error handling and special values

Errors are reported consistent with math_errhandling

If the implementation supports IEEE floating-point arithmetic,

- cacos( conj (z)) == conj (cacos(z))
- If `z` is `±0+0i`, the result is `π/2-0i`
- If `z` is `±0+NaNi`, the result is `π/2+NaNi`
- If `z` is `x+∞i` (for any finite x), the result is `π/2-∞i`
- If `z` is `x+NaNi` (for any nonzero finite x), the result is `NaN+NaNi` and FE_INVALID may be raised.
- If `z` is `-∞+yi` (for any positive finite y), the result is `π-∞i`
- If `z` is `-∞+yi` (for any positive finite y), the result is `+0-∞i`
- If `z` is `-∞+∞i`, the result is `3π/4-∞i`
- If `z` is `+∞+∞i`, the result is `π/4-∞i`
- If `z` is `±∞+NaNi`, the result is `NaN±∞i` (the sign of the imaginary part is unspecified)
- If `z` is `NaN+yi` (for any finite y), the result is `NaN+NaNi` and FE_INVALID may be raised
- If `z` is `NaN+∞i`, the result is `NaN-∞i`
- If `z` is `NaN+NaNi`, the result is `NaN+NaNi`

### Notes

Inverse cosine (or arc cosine) is a multivalued function and requires a branch cut on the complex plane. The branch cut is conventially placed at the line segments (-∞,-1) and (1,∞) of the real axis.

The mathematical definition of the principal value of arc cosine is acos z =

1
2

π + iln(iz + √1-z2
)

For any z, acos(z) = π - acos(-z)

### Example

Run this code

    #include <stdio.h>
    #include <math.h>
    #include <complex.h>

    int main(void)
    {
        double complex z = cacos(-2);
        printf("cacos(-2+0i) = %f%+fi\n", creal(z), cimag(z));

        double complex z2 = cacos(conj(-2)); // or CMPLX(-2, -0.0)
        printf("cacos(-2-0i) (the other side of the cut) = %f%+fi\n", creal(z2), cimag(z2));

        // for any z, acos(z) = pi - acos(-z)
        double pi = acos(-1);
        double complex z3 = ccos(pi-z2);
        printf("ccos(pi - cacos(-2-0i) = %f%+fi\n", creal(z3), cimag(z3));
    }

Output:

    cacos(-2+0i) = 3.141593-1.316958i
    cacos(-2-0i) (the other side of the cut) = 3.141593+1.316958i
    ccos(pi - cacos(-2-0i) = 2.000000+0.000000i

### References

- C11 standard (ISO/IEC 9899:2011):

  - 7.3.5.1 The cacos functions (p: 190)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - G.6.1.1 The cacos functions (p: 539)

  - G.7 Type-generic math <tgmath.h> (p: 545)

- C99 standard (ISO/IEC 9899:1999):

  - 7.3.5.1 The cacos functions (p: 172)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - G.6.1.1 The cacos functions (p: 474)

  - G.7 Type-generic math <tgmath.h> (p: 480)

### See also

casincasinfcasinl (C99)(C99)(C99) | computes the complex arc sine (function)
catancatanfcatanl (C99)(C99)(C99) | computes the complex arc tangent (function)
ccosccosfccosl (C99)(C99)(C99) | computes the complex cosine (function)
acosacosfacosl (C99)(C99) | computes arc cosine (arccos(x)) (function)

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