sin()

transform: translateX(calc(cos(0deg) * 140px))
  translateY(calc(sin(0deg) * -140px));

transform: translateX(calc(cos(90deg) * 140px))
  translateY(calc(sin(90deg) * -140px));

transform: translateX(calc(cos(135deg) * 140px))
  translateY(calc(sin(135deg) * -140px));

transform: translateX(calc(cos(180deg) * 140px))
  translateY(calc(sin(180deg) * -140px));

transform: translateX(calc(cos(-45deg) * 140px))
  translateY(calc(sin(-45deg) * -140px));

<div class="circle">
  <span class="dot" id="example-element"></span>
</div>

:root {
  --radius: 140px;
  --dot-size: 10px;
}
.circle {
  display: grid;
  place-content: center;
  margin: 0 auto;
  width: calc(var(--radius) * 2);
  aspect-ratio: 1;
  border-radius: 50%;
  border: 2px solid #666666;
  background-image:
    radial-gradient(black var(--dot-size), transparent var(--dot-size)),
    linear-gradient(135deg, blue, deepskyblue, lightgreen, lavender, honeydew);
}
.dot {
  display: block;
  width: var(--dot-size);
  aspect-ratio: 1;
  border-radius: 50%;
  border: 2px solid #666666;
  background-color: #ff6666;
  transform: translateX(calc(cos(0deg) * var(--radius)))
    translateY(calc(sin(0deg) * var(--radius) * -1));
}

/* 単一の <angle> 値 */
width: calc(100px * sin(45deg));
width: calc(100px * sin(0.25turn));
width: calc(100px * sin(1.0471967rad));
/* 単一の <number> 値 */
width: calc(100px * sin(63.673));
width: calc(100px * sin(2 * 0.125));
/* その他の値 */
width: calc(100px * sin(pi / 2));
width: calc(100px * sin(e / 4));

div {
  background-color: red;
  width: calc(sin(30deg) * 100px);
  height: calc(sin(30deg) * 100px);
}

div {
  animation-name: myAnimation;
  animation-duration: calc(sin(0.25turn) * 1s);
}