Sine, Cosine, and Tangent
Introduced| English |
|---|
| sine/saɪn/ |
| cosine/ˈkəʊsaɪn/ |
| tangent/ˈtændʒənt/ |
| unit circle/ˈjuːnɪt ˈsɜːkl/ |
| radians/ˈreɪdɪənz/ |
Three ratios that unlock circles
- Long ago, surveyors and astronomers needed to link angles to lengths.
- The answer was three ratios inside a right triangle: sine, cosine, and tangent.
- They turn an angle into a number, and a number back into an angle.
- These three functions are the doorway to everything periodic.
Right-triangle ratios
- For an acute angle in a right triangle, label the sides opposite, adjacent, and hypotenuse.
- Sine 正弦: $\sin\theta = \dfrac{\text{opposite}}{\text{hypotenuse}}$.
- Cosine 余弦: $\cos\theta = \dfrac{\text{adjacent}}{\text{hypotenuse}}$.
- Tangent 正切: $\tan\theta = \dfrac{\text{opposite}}{\text{adjacent}}$. Remember SOH-CAH-TOA.
In a right triangle, $\sin\theta$ equals…
SOH: sine is opposite over hypotenuse. CAH: cosine is adjacent over hypotenuse. TOA: tangent is opposite over adjacent.
The unit circle
- Draw a circle of radius $1$ centred at the origin — the unit circle 单位圆.
- A point on it at angle $\theta$ has coordinates exactly $(\cos\theta, \sin\theta)$.
- So cosine is the x-coordinate and sine is the y-coordinate.
- This lets angles beyond $90°$ have sine and cosine too — all the way around.
Spin a point around the unit circle
The x-coordinate of the point is the cosine of the angle, and the y-coordinate is the sine.
On the unit circle, a point at angle $\theta$ has coordinates…
On a circle of radius 1, the point is $(\cos\theta, \sin\theta)$ — cosine is the x-coordinate, sine the y.
Select all true statements.
A full circle is $360°$ (or $2\pi$ radians), not $100°$. The other three are correct.
Radians and tangent
- Angles can be measured in degrees or in radians 弧度, where a full circle is $2\pi$.
- Radians measure the arc length swept on the unit circle, so $180° = \pi$.
- Tangent connects the other two: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
- On the unit circle, tangent is the slope of the line from the origin to the point.
Tangent equals sine divided by ____ (so $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$).
The tangent is sine divided by cosine: $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
Angles can be measured in radians, where a full circle is $2\pi$.
A full turn is $360°$ or, in radians, $2\pi$. Radians measure the arc length on the unit circle.
Around and around
- As the point travels once around the circle, sine and cosine cycle through all their values.
- Past one full turn, everything repeats — which is why these functions are periodic.
- Cosine starts at $1$ (angle $0$); sine starts at $0$.
- The unit circle is the single picture behind every trig value.
Keep track of your angle units. $\sin(90)$ is nearly $1$ if $90$ means degrees, but a very different value if $90$ means radians. Always know whether a problem is in degrees or radians before you compute.
Find $\sin\theta$ and $\cos\theta$ for the angle whose unit-circle point is $(0.6, 0.8)$.
- The x-coordinate is cosine: $\cos\theta = 0.6$.
- The y-coordinate is sine: $\sin\theta = 0.8$.
- Check: $0.6^2 + 0.8^2 = 0.36 + 0.64 = 1$, as the unit circle requires.
Sine, cosine, and tangent are ratios in a right triangle (SOH-CAH-TOA). On the unit circle, a point at angle $\theta$ is $(\cos\theta, \sin\theta)$, which extends the functions to any angle. Angles measure in degrees or radians ($2\pi$ per turn), and $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$.