Sine and Cosine Function Values
Introduced| English |
|---|
| sine/saɪn/ |
| cosine/ˈkəʊsaɪn/ |
| unit circle/ˈjuːnɪt ˈsɜːkl/ |
| radians/ˈreɪdɪənz/ |
| period/ˈpɪərɪəd/ |
Values you can read off a circle
- Every angle has a sine and a cosine — but where do the numbers come from?
- They are just coordinates on the unit circle, read straight off.
- A handful of "special" angles give clean, memorable values.
- Learn those, and the symmetry of the circle hands you the rest.
Special-angle values
- At $\theta = 0$: the point is $(1, 0)$, so $\cos 0 = 1$ and $\sin 0 = 0$.
- At $\theta = \tfrac{\pi}{2}$ ($90°$): the point is $(0, 1)$, so $\cos = 0$, $\sin = 1$.
- The angles $30°, 45°, 60°$ give the familiar values with $\tfrac12$, $\tfrac{\sqrt2}{2}$, $\tfrac{\sqrt3}{2}$.
- Reading a sine 正弦 or cosine 余弦 is just reading a coordinate.
Read sine and cosine at any angle
Move the angle and watch the sine (height) and cosine (across) values change and repeat.
What is $\sin 0$?
At angle $0$, the unit-circle point is $(1, 0)$, so the y-coordinate $\sin 0 = 0$.
What is $\cos 0$?
At angle $0$, the point is $(1, 0)$, so the x-coordinate $\cos 0 = 1$.
Signs by quadrant
- In the top-right quadrant both coordinates are positive, so sine and cosine are positive.
- Cross into other quadrants and one or both turn negative.
- "All, Sine, Tangent, Cosine" (ASTC) records which are positive, quadrant by quadrant.
- The unit circle 单位圆 makes every sign obvious — just look at the coordinate.
The output of $\sin\theta$ always lies between…
On the unit circle the y-coordinate never leaves $[-1, 1]$, so $-1 \le \sin\theta \le 1$.
Select all true statements about sine and cosine values.
Sine never reaches $2$ — it is capped at $1$. The other three are correct.
Repeating every full turn
- After one full revolution the point lands exactly where it began.
- So the values repeat every $2\pi$ radians 弧度 (or $360°$).
- That regular repeat is the period 周期 of sine and cosine.
- Adding any multiple of $2\pi$ to an angle leaves its sine and cosine unchanged.
Sine and cosine values repeat every ____ radians (one full turn).
After a full revolution of $2\pi$ (or $360°$), the point returns to where it started, so the values repeat.
Bounded outputs
- Because the circle has radius $1$, no coordinate ever exceeds $1$ or drops below $-1$.
- So both sine and cosine always satisfy $-1 \le \text{value} \le 1$.
- The maximum, $1$, and minimum, $-1$, occur at the top/bottom or left/right of the circle.
- This bound is why sinusoidal waves have a fixed height.
Sine and cosine can never be $2$ or $-5$. Their outputs are locked into $[-1, 1]$ by the unit circle. If a calculation gives $\sin\theta = 1.4$, you have made an error somewhere.
Find $\sin\tfrac{\pi}{2}$, $\cos\pi$, and $\sin 3\pi$.
- $\sin\tfrac{\pi}{2}$: the top of the circle $(0,1)$, so $\sin = 1$.
- $\cos\pi$: the left point $(-1, 0)$, so $\cos = -1$.
- $\sin 3\pi = \sin(\pi + 2\pi) = \sin\pi = 0$ — the extra $2\pi$ changes nothing.
Sine and cosine values are coordinates on the unit circle. Special angles give clean values, and the quadrant fixes the sign. The values repeat every $2\pi$ radians (the period), and both always stay within $[-1, 1]$.