diff --git a/doc/source/fmprb.rst b/doc/source/fmprb.rst
index 7f451365..6e6f9907 100644
--- a/doc/source/fmprb.rst
+++ b/doc/source/fmprb.rst
@@ -480,6 +480,24 @@ Elementary functions
 
     Sets `s = \sin \pi x`, `c = \cos \pi x`.
 
+.. function:: void fmprb_sin_pi_fmpq(fmprb_t s, const fmpq_t x, long prec)
+
+.. function:: void fmprb_cos_pi_fmpq(fmprb_t c, const fmpq_t x, long prec)
+
+.. function:: void fmprb_sin_cos_pi_fmpq(fmprb_t s, fmprb_t c, const fmpq_t x, long prec)
+
+    Sets `s = \sin \pi x`, `c = \cos \pi x` where `x` is a rational
+    number (whose numerator and denominator are assumed to be reduced).
+    We first use trigonometric symmetries to reduce the argument to the
+    octant `[0, 1/4]`. If the reduced argument is one of
+    `0, 1/4, 1/6, 1/8, 1/5, 1/10`, we use an explicit algebraic
+    formula. Otherwise
+    we multiply by a numerical approximation of `\pi`
+    and evaluate the trigonometric function the usual way.
+    Since the argument has been reduced to the first octant, this
+    gives full accuracy even if the original argument is close to
+    some root other the origin.
+
 .. function:: void fmprb_atan(fmprb_t z, const fmprb_t x, long prec)
 
     Sets `z = \tan^{-1} x`. Letting `d = \max(0, |m| - r)`,
diff --git a/fmprb.h b/fmprb.h
index afbb88e7..38b35b56 100644
--- a/fmprb.h
+++ b/fmprb.h
@@ -275,6 +275,10 @@ void fmprb_sin_pi(fmprb_t s, const fmprb_t x, long prec);
 void fmprb_cos_pi(fmprb_t c, const fmprb_t x, long prec);
 void fmprb_sin_cos_pi(fmprb_t s, fmprb_t c, const fmprb_t x, long prec);
 
+void fmprb_sin_cos_pi_fmpq(fmprb_t s, fmprb_t c, const fmpq_t x, long prec);
+void fmprb_sin_pi_fmpq(fmprb_t s, const fmpq_t x, long prec);
+void fmprb_cos_pi_fmpq(fmprb_t c, const fmpq_t x, long prec);
+
 void fmprb_sinh(fmprb_t z, const fmprb_t x, long prec);
 void fmprb_cosh(fmprb_t z, const fmprb_t x, long prec);
 void fmprb_sinh_cosh(fmprb_t s, fmprb_t c, const fmprb_t x, long prec);
diff --git a/fmprb/sin_cos_pi_fmpq.c b/fmprb/sin_cos_pi_fmpq.c
new file mode 100644
index 00000000..942832e9
--- /dev/null
+++ b/fmprb/sin_cos_pi_fmpq.c
@@ -0,0 +1,289 @@
+/*=============================================================================
+
+    This file is part of ARB.
+
+    ARB is free software; you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation; either version 2 of the License, or
+    (at your option) any later version.
+
+    ARB is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with ARB; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
+
+=============================================================================*/
+/******************************************************************************
+
+    Copyright (C) 2012 Fredrik Johansson
+
+******************************************************************************/
+
+#include "fmprb.h"
+
+/* evaluates sin(pi v/w), cos(pi v/w) assuming
+   (for best performance and accuracy)
+   v, w reduced and 0 <= v/w <= 1/4. */
+
+void
+_fmprb_sin_cos_pi_fmpq_oct(fmprb_t s, fmprb_t c,
+            const fmpz_t v, const fmpz_t w, long prec)
+{
+    if (*v == 0)
+    {
+        fmprb_zero(s);
+        fmprb_one(c);
+    }
+    else if (*v == 1 && *w == 4)
+    {
+        fmprb_sqrt_ui(s, 2, prec);
+        fmprb_mul_2exp_si(s, s, -1);
+        fmprb_set(c, s);
+    }
+    else if (*v == 1 && *w == 6)
+    {
+        fmprb_one(s);
+        fmprb_mul_2exp_si(s, s, -1);
+        fmprb_sqrt_ui(c, 3, prec);
+        fmprb_mul_2exp_si(c, c, -1);
+    }
+    else if (*v == 1 && *w == 8)
+    {
+        fmprb_sqrt_ui(s, 2, prec);
+        fmprb_add_ui(c, s, 2, prec);
+        fmprb_sqrt(c, c, prec);
+        fmprb_mul_2exp_si(c, c, -1);
+        fmprb_sub_ui(s, s, 2, prec);
+        fmprb_neg(s, s);
+        fmprb_sqrt(s, s, prec);
+        fmprb_mul_2exp_si(s, s, -1);
+    }
+    else if (*v == 1 && *w == 5)
+    {
+        fmprb_sqrt_ui(s, 5, prec);
+        fmprb_add_ui(c, s, 1, prec);
+        fmprb_mul_2exp_si(c, c, -2);
+        fmprb_sub_ui(s, s, 5, prec);
+        fmprb_neg(s, s);
+        fmprb_mul_2exp_si(s, s, -3);
+        fmprb_sqrt(s, s, prec);
+    }
+    else if (*v == 1 && *w == 10)
+    {
+        fmprb_sqrt_ui(c, 5, prec);
+        fmprb_sub_ui(s, c, 1, prec);
+        fmprb_mul_2exp_si(s, s, -2);
+        fmprb_add_ui(c, c, 5, prec);
+        fmprb_mul_2exp_si(c, c, -3);
+        fmprb_sqrt(c, c, prec);
+    }
+    else
+    {
+        fmprb_const_pi(s, prec);
+        fmprb_mul_fmpz(s, s, v, prec);
+        fmprb_div_fmpz(s, s, w, prec);
+        fmprb_sin_cos(s, c, s, prec);
+    }
+}
+
+void
+_fmprb_sin_pi_fmpq_oct(fmprb_t s, const fmpz_t v, const fmpz_t w, long prec)
+{
+    if (*v == 0)
+    {
+        fmprb_zero(s);
+    }
+    else if (*v == 1 && *w == 4)
+    {
+        fmprb_sqrt_ui(s, 2, prec);
+        fmprb_mul_2exp_si(s, s, -1);
+    }
+    else if (*v == 1 && *w == 6)
+    {
+        fmprb_one(s);
+        fmprb_mul_2exp_si(s, s, -1);
+    }
+    else if (*v == 1 && *w == 8)
+    {
+        fmprb_sqrt_ui(s, 2, prec);
+        fmprb_sub_ui(s, s, 2, prec);
+        fmprb_neg(s, s);
+        fmprb_sqrt(s, s, prec);
+        fmprb_mul_2exp_si(s, s, -1);
+    }
+    else if (*v == 1 && *w == 5)
+    {
+        fmprb_sqrt_ui(s, 5, prec);
+        fmprb_sub_ui(s, s, 5, prec);
+        fmprb_neg(s, s);
+        fmprb_mul_2exp_si(s, s, -3);
+        fmprb_sqrt(s, s, prec);
+    }
+    else if (*v == 1 && *w == 10)
+    {
+        fmprb_sqrt_ui(s, 5, prec);
+        fmprb_sub_ui(s, s, 1, prec);
+        fmprb_mul_2exp_si(s, s, -2);
+    }
+    else
+    {
+        fmprb_const_pi(s, prec);
+        fmprb_mul_fmpz(s, s, v, prec);
+        fmprb_div_fmpz(s, s, w, prec);
+        fmprb_sin(s, s, prec);
+    }
+}
+
+void
+_fmprb_cos_pi_fmpq_oct(fmprb_t c, const fmpz_t v, const fmpz_t w, long prec)
+{
+    if (*v == 0)
+    {
+        fmprb_one(c);
+    }
+    else if (*v == 1 && *w == 4)
+    {
+        fmprb_sqrt_ui(c, 2, prec);
+        fmprb_mul_2exp_si(c, c, -1);
+    }
+    else if (*v == 1 && *w == 6)
+    {
+        fmprb_sqrt_ui(c, 3, prec);
+        fmprb_mul_2exp_si(c, c, -1);
+    }
+    else if (*v == 1 && *w == 8)
+    {
+        fmprb_sqrt_ui(c, 2, prec);
+        fmprb_add_ui(c, c, 2, prec);
+        fmprb_sqrt(c, c, prec);
+        fmprb_mul_2exp_si(c, c, -1);
+    }
+    else if (*v == 1 && *w == 5)
+    {
+        fmprb_sqrt_ui(c, 5, prec);
+        fmprb_add_ui(c, c, 1, prec);
+        fmprb_mul_2exp_si(c, c, -2);
+    }
+    else if (*v == 1 && *w == 10)
+    {
+        fmprb_sqrt_ui(c, 5, prec);
+        fmprb_add_ui(c, c, 5, prec);
+        fmprb_mul_2exp_si(c, c, -3);
+        fmprb_sqrt(c, c, prec);
+    }
+    else
+    {
+        fmprb_const_pi(c, prec);
+        fmprb_mul_fmpz(c, c, v, prec);
+        fmprb_div_fmpz(c, c, w, prec);
+        fmprb_cos(c, c, prec);
+    }
+}
+
+unsigned int
+reduce_octant(fmpz_t v, fmpz_t w, const fmpq_t x)
+{
+    const fmpz * p = fmpq_numref(x);
+    const fmpz * q = fmpq_denref(x);
+    fmpz_t u;
+    unsigned int octant;
+    mp_bitcnt_t vval, wval;
+
+    fmpz_init(u);
+
+    fmpz_mul_2exp(w, p, 2);
+    fmpz_fdiv_qr(w, v, w, q);
+    octant = fmpz_fdiv_ui(w, 8);
+    fmpz_mul_2exp(w, q, 2);
+
+    if (octant % 2 != 0)
+        fmpz_sub(v, q, v);
+
+    vval = fmpz_val2(v);
+    wval = fmpz_val2(w);
+    vval = FLINT_MIN(vval, wval);
+
+    if (vval != 0)
+    {
+        fmpz_tdiv_q_2exp(v, v, vval);
+        fmpz_tdiv_q_2exp(w, w, vval);
+    }
+
+    fmpz_clear(u);
+
+    return octant;
+}
+
+void
+fmprb_sin_cos_pi_fmpq(fmprb_t s, fmprb_t c, const fmpq_t x, long prec)
+{
+    fmpz_t v, w;
+    unsigned int octant;
+
+    fmpz_init(v);
+    fmpz_init(w);
+
+    octant = reduce_octant(v, w, x);
+
+    if ((octant + 1) % 4 < 2)
+        _fmprb_sin_cos_pi_fmpq_oct(s, c, v, w, prec);
+    else
+        _fmprb_sin_cos_pi_fmpq_oct(c, s, v, w, prec);
+
+    if ((octant + 6) % 8 < 4)  fmprb_neg(c, c);
+    if (octant >= 4)           fmprb_neg(s, s);
+
+    fmpz_clear(v);
+    fmpz_clear(w);
+}
+
+void
+fmprb_sin_pi_fmpq(fmprb_t s, const fmpq_t x, long prec)
+{
+    fmpz_t v, w;
+    unsigned int octant;
+
+    fmpz_init(v);
+    fmpz_init(w);
+
+    octant = reduce_octant(v, w, x);
+
+    if ((octant + 1) % 4 < 2)
+        _fmprb_sin_pi_fmpq_oct(s, v, w, prec);
+    else
+        _fmprb_cos_pi_fmpq_oct(s, v, w, prec);
+
+    if (octant >= 4)
+        fmprb_neg(s, s);
+
+    fmpz_clear(v);
+    fmpz_clear(w);
+}
+
+void
+fmprb_cos_pi_fmpq(fmprb_t c, const fmpq_t x, long prec)
+{
+    fmpz_t v, w;
+    unsigned int octant;
+
+    fmpz_init(v);
+    fmpz_init(w);
+
+    octant = reduce_octant(v, w, x);
+
+    if ((octant + 1) % 4 < 2)
+        _fmprb_cos_pi_fmpq_oct(c, v, w, prec);
+    else
+        _fmprb_sin_pi_fmpq_oct(c, v, w, prec);
+
+    if ((octant + 6) % 8 < 4)
+        fmprb_neg(c, c);
+
+    fmpz_clear(v);
+    fmpz_clear(w);
+}
+
diff --git a/fmprb/test/t-cos_pi_fmpq.c b/fmprb/test/t-cos_pi_fmpq.c
new file mode 100644
index 00000000..5c0de92b
--- /dev/null
+++ b/fmprb/test/t-cos_pi_fmpq.c
@@ -0,0 +1,78 @@
+/*=============================================================================
+
+    This file is part of ARB.
+
+    ARB is free software; you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation; either version 2 of the License, or
+    (at your option) any later version.
+
+    ARB is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with ARB; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
+
+=============================================================================*/
+/******************************************************************************
+
+    Copyright (C) 2013 Fredrik Johansson
+
+******************************************************************************/
+
+#include "fmprb.h"
+
+int main()
+{
+    long iter;
+    flint_rand_t state;
+
+    printf("cos_pi_fmpq....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+    for (iter = 0; iter < 10000; iter++)
+    {
+        fmprb_t c1, c2;
+        fmpq_t x;
+        long prec;
+
+        prec = 2 + n_randint(state, 1000);
+
+        fmprb_init(c1);
+        fmprb_init(c2);
+        fmpq_init(x);
+
+        fmpq_randtest(x, state, 1 + n_randint(state, 200));
+
+        fmprb_cos_pi_fmpq(c1, x, prec);
+
+        fmprb_const_pi(c2, prec);
+        fmprb_mul_fmpz(c2, c2, fmpq_numref(x), prec);
+        fmprb_div_fmpz(c2, c2, fmpq_denref(x), prec);
+        fmprb_cos(c2, c2, prec);
+
+        if (!fmprb_overlaps(c1, c2))
+        {
+            printf("FAIL: overlap\n\n");
+            printf("x = "); fmpq_print(x); printf("\n\n");
+            printf("c1 = "); fmprb_printd(c1, 15); printf("\n\n");
+            printf("c2 = "); fmprb_printd(c2, 15); printf("\n\n");
+            abort();
+        }
+
+        fmprb_clear(c1);
+        fmprb_clear(c2);
+        fmpq_clear(x);
+    }
+
+    flint_randclear(state);
+    _fmpz_cleanup();
+    printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+
diff --git a/fmprb/test/t-sin_cos_pi_fmpq.c b/fmprb/test/t-sin_cos_pi_fmpq.c
new file mode 100644
index 00000000..3a53875d
--- /dev/null
+++ b/fmprb/test/t-sin_cos_pi_fmpq.c
@@ -0,0 +1,84 @@
+/*=============================================================================
+
+    This file is part of ARB.
+
+    ARB is free software; you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation; either version 2 of the License, or
+    (at your option) any later version.
+
+    ARB is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with ARB; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
+
+=============================================================================*/
+/******************************************************************************
+
+    Copyright (C) 2013 Fredrik Johansson
+
+******************************************************************************/
+
+#include "fmprb.h"
+
+int main()
+{
+    long iter;
+    flint_rand_t state;
+
+    printf("sin_cos_pi_fmpq....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+    for (iter = 0; iter < 10000; iter++)
+    {
+        fmprb_t s1, c1, s2, c2;
+        fmpq_t x;
+        long prec;
+
+        prec = 2 + n_randint(state, 1000);
+
+        fmprb_init(s1);
+        fmprb_init(c1);
+        fmprb_init(s2);
+        fmprb_init(c2);
+        fmpq_init(x);
+
+        fmpq_randtest(x, state, 1 + n_randint(state, 200));
+
+        fmprb_sin_cos_pi_fmpq(s1, c1, x, prec);
+
+        fmprb_const_pi(s2, prec);
+        fmprb_mul_fmpz(s2, s2, fmpq_numref(x), prec);
+        fmprb_div_fmpz(s2, s2, fmpq_denref(x), prec);
+        fmprb_sin_cos(s2, c2, s2, prec);
+
+        if (!fmprb_overlaps(s1, s2) || !fmprb_overlaps(c1, c2))
+        {
+            printf("FAIL: overlap\n\n");
+            printf("x = "); fmpq_print(x); printf("\n\n");
+            printf("s1 = "); fmprb_printd(s1, 15); printf("\n\n");
+            printf("c1 = "); fmprb_printd(c1, 15); printf("\n\n");
+            printf("s2 = "); fmprb_printd(s2, 15); printf("\n\n");
+            printf("c2 = "); fmprb_printd(c2, 15); printf("\n\n");
+            abort();
+        }
+
+        fmprb_clear(s1);
+        fmprb_clear(c1);
+        fmprb_clear(s2);
+        fmprb_clear(c2);
+        fmpq_clear(x);
+    }
+
+    flint_randclear(state);
+    _fmpz_cleanup();
+    printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+
diff --git a/fmprb/test/t-sin_pi_fmpq.c b/fmprb/test/t-sin_pi_fmpq.c
new file mode 100644
index 00000000..0b418d8f
--- /dev/null
+++ b/fmprb/test/t-sin_pi_fmpq.c
@@ -0,0 +1,78 @@
+/*=============================================================================
+
+    This file is part of ARB.
+
+    ARB is free software; you can redistribute it and/or modify
+    it under the terms of the GNU General Public License as published by
+    the Free Software Foundation; either version 2 of the License, or
+    (at your option) any later version.
+
+    ARB is distributed in the hope that it will be useful,
+    but WITHOUT ANY WARRANTY; without even the implied warranty of
+    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+    GNU General Public License for more details.
+
+    You should have received a copy of the GNU General Public License
+    along with ARB; if not, write to the Free Software
+    Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
+
+=============================================================================*/
+/******************************************************************************
+
+    Copyright (C) 2013 Fredrik Johansson
+
+******************************************************************************/
+
+#include "fmprb.h"
+
+int main()
+{
+    long iter;
+    flint_rand_t state;
+
+    printf("sin_pi_fmpq....");
+    fflush(stdout);
+
+    flint_randinit(state);
+
+    for (iter = 0; iter < 10000; iter++)
+    {
+        fmprb_t s1, s2;
+        fmpq_t x;
+        long prec;
+
+        prec = 2 + n_randint(state, 1000);
+
+        fmprb_init(s1);
+        fmprb_init(s2);
+        fmpq_init(x);
+
+        fmpq_randtest(x, state, 1 + n_randint(state, 200));
+
+        fmprb_sin_pi_fmpq(s1, x, prec);
+
+        fmprb_const_pi(s2, prec);
+        fmprb_mul_fmpz(s2, s2, fmpq_numref(x), prec);
+        fmprb_div_fmpz(s2, s2, fmpq_denref(x), prec);
+        fmprb_sin(s2, s2, prec);
+
+        if (!fmprb_overlaps(s1, s2))
+        {
+            printf("FAIL: overlap\n\n");
+            printf("x = "); fmpq_print(x); printf("\n\n");
+            printf("s1 = "); fmprb_printd(s1, 15); printf("\n\n");
+            printf("s2 = "); fmprb_printd(s2, 15); printf("\n\n");
+            abort();
+        }
+
+        fmprb_clear(s1);
+        fmprb_clear(s2);
+        fmpq_clear(x);
+    }
+
+    flint_randclear(state);
+    _fmpz_cleanup();
+    printf("PASS\n");
+    return EXIT_SUCCESS;
+}
+