Sine and Cosine Function Values
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| sine/saɪn/ | hàm sin |
| cosine/ˈkəʊsaɪn/ | hàm cosin |
| unit circle/ˈjuːnɪt ˈsɜːkl/ | đơn vị tròn |
| radians/ˈreɪdɪənz/ | radian |
| period/ˈpɪərɪəd/ | chu kỳ |
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 $\sin 0$?
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At angle $0$, the unit-circle point is $(1, 0)$, so the y-coordinate $\sin 0 = 0$.
What is $\cos 0$?
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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…
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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.
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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).
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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 $[-1, 1]$.
Write from a sine of 1
- Read sine and cosine at a quarter turn.
- The lesson reads them at angle 0.
What is sine of pi over 2?
The top of the unit circle has height 1.
What is sine of pi over 2?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Sine reads the y-coordinate. Other options — 0 (the lesson value of sine of 0); 0 (the cosine of pi over 2).