Equivalent Trigonometric Expressions · 三角表达式的等价形式
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| identity/aɪˈdentɪti/ | 恒等式 | héng děng shì |
| equivalent/ɪˈkwɪvələnt/ | 等价 | děng jià |
| Pythagorean identity/ˈpaɪθəɡɔːrɪən aɪˈdentɪti/ | 勾股恒等式 | gōu gǔ héng děng shì |
| sine/saɪn/ | 正弦 | zhèng xián |
| cosine/ˈkəʊsaɪn/ | 余弦 | yú xián |
Two faces of the same expression
- The same trig quantity can be written in many equal-looking-but-different ways.
- $\sin^2\theta$ can become $1 - \cos^2\theta$; $\tan\theta$ can become $\tfrac{\sin\theta}{\cos\theta}$.
- These swaps are powered by identities — equations true for every angle.
- Choosing the right form turns a hard problem into an easy one.
同一个表达式的两副面孔
- 同一个三角量可以写成许多看起来不同、却相等的形式。
- $\sin^2\theta$ 可以变成 $1 - \cos^2\theta$;$\tan\theta$ 可以变成 $\tfrac{\sin\theta}{\cos\theta}$。
- 这些替换由恒等式驱动——对每一个角都成立的等式。
- 选对形式,能把难题变成易题。
What an identity is
- A trigonometric identity 恒等式 is an equation true for all valid angles.
- This is different from an equation you solve, which holds only at special angles.
- Two expressions linked by an identity are equivalent 等价 everywhere.
- So you can substitute one for the other whenever it helps.
什么是恒等式
- 一个三角恒等式(identity)是对所有有效角都成立的等式。
- 这与你求解的方程不同,后者只在特殊角处成立。
- 由恒等式联系起来的两个表达式,处处等价(equivalent)。
- 所以只要有帮助,你就可以用一个替换另一个。
Watch the Pythagorean identity hold · 看勾股恒等式成立
For any angle, the point on the unit circle satisfies cos squared plus sin squared equals 1 — the master trig identity. · 对任何角,单位圆上的点都满足余弦平方加正弦平方等于 1——最核心的三角恒等式。
A trigonometric identity is an equation that is true… · 一个三角恒等式是一个……都成立的等式。
An identity holds for all · 所有 valid inputs — unlike an equation you solve for particular values. · 恒等式对所有有效输入都成立——与你为特定值求解的方程不同。
The Pythagorean identity
- The master identity is the Pythagorean identity 勾股恒等式: $\sin^2\theta + \cos^2\theta = 1$.
- It comes straight from the unit circle, where $(\cos\theta, \sin\theta)$ has radius $1$.
- Rearranged, it gives $\cos^2\theta = 1 - \sin^2\theta$ and $\sin^2\theta = 1 - \cos^2\theta$.
- Dividing by $\cos^2$ or $\sin^2$ produces the secant and cosecant versions.
勾股恒等式
- 最核心的恒等式是勾股恒等式(Pythagorean identity):$\sin^2\theta + \cos^2\theta = 1$。
- 它直接来自单位圆,那里 $(\cos\theta, \sin\theta)$ 的半径是 $1$。
- 变形后,它给出 $\cos^2\theta = 1 - \sin^2\theta$ 和 $\sin^2\theta = 1 - \cos^2\theta$。
- 除以 $\cos^2$ 或 $\sin^2$,就产生正割和余割的版本。
The Pythagorean identity states that… · 勾股恒等式说……
Because a unit-circle point $(\cos\theta, \sin\theta)$ has radius $1$: $\cos^2\theta + \sin^2\theta = 1$. · 因为单位圆上的点 $(\cos\theta, \sin\theta)$ 半径为 $1$:$\cos^2\theta + \sin^2\theta = 1$。
Select all · 所有 true statements about trig identities. · 选出关于三角恒等式的所有正确说法。
An identity is true for all · 所有 angles, not just one. The other three are correct. · 恒等式对所有角都成立,而不只是一个。其余三条正确。
A toolbox of identities
- Reciprocal identities: $\sec = \tfrac{1}{\cos}$, and so on.
- Quotient identity: $\tan\theta = \tfrac{\sin\theta}{\cos\theta}$.
- Even/odd: $\sin(-\theta) = -\sin\theta$, $\cos(-\theta) = \cos\theta$.
- Together they let you rewrite almost any trig expression in a new form.
一整套恒等式工具
- 倒数恒等式:$\sec = \tfrac{1}{\cos}$,等等。
- 商恒等式:$\tan\theta = \tfrac{\sin\theta}{\cos\theta}$。
- 奇偶性:$\sin(-\theta) = -\sin\theta$,$\cos(-\theta) = \cos\theta$。
- 它们合在一起,让你几乎能把任何三角表达式改写成新形式。
If $\sin\theta = 0.6$, then from the Pythagorean identity $\cos^2\theta = 1 - 0.36 =$ ____. · 若 $\sin\theta = 0.6$,则由勾股恒等式 $\cos^2\theta = 1 - 0.36 =$ ____。
$\cos^2\theta = 1 - \sin^2\theta = 1 - 0.36 = 0.64$, so $\cos\theta = \pm 0.8$. · $\cos^2\theta = 1 - \sin^2\theta = 1 - 0.36 = 0.64$,所以 $\cos\theta = \pm 0.8$。
Identities let you rewrite a trig expression as an equivalent one that may be easier to work with. · 恒等式让你把一个三角表达式改写成一个可能更好处理的等价式子。
That is their main use: swap in an equal form to simplify an expression or solve an equation. · 这正是它们的主要用途:换成一个相等的形式来化简表达式或求解方程。
Using identities
- To simplify, replace part of an expression with an equal form until it collapses.
- To solve an equation, rewrite everything in terms of one function (say, all sine).
- To verify an identity, transform one side until it matches the other.
- The sine 正弦 and cosine 余弦 forms are usually the safest to aim for.
使用恒等式
- 要化简,就把表达式的一部分换成相等的形式,直到它塌缩。
- 要解方程,就把一切都改写成一个函数(比如全用正弦)。
- 要验证一个恒等式,就变形一边直到它与另一边相符。
- 通常最保险的目标是正弦(sine)和余弦(cosine)的形式。
An identity holds for all angles, so both sides must match everywhere — you cannot "prove" one by plugging in a single value. Testing $\theta = 0$ can disprove a false identity, but confirming one angle never proves a true one.
恒等式对所有角都成立,所以两边必须处处相符——你不能靠代入单独一个值来"证明"它。测试 $\theta = 0$ 可以推翻一个错误的恒等式,但确认一个角永远不能证明一个正确的恒等式。
Given $\sin\theta = 0.6$ with $\theta$ acute, find $\cos\theta$.
- Pythagorean identity: $\cos^2\theta = 1 - \sin^2\theta = 1 - 0.36 = 0.64$.
- So $\cos\theta = \pm\sqrt{0.64} = \pm 0.8$.
- Since $\theta$ is acute, cosine is positive: $\cos\theta = 0.8$.
已知 $\sin\theta = 0.6$ 且 $\theta$ 为锐角,求 $\cos\theta$。
- 勾股恒等式:$\cos^2\theta = 1 - \sin^2\theta = 1 - 0.36 = 0.64$。
- 所以 $\cos\theta = \pm\sqrt{0.64} = \pm 0.8$。
- 因为 $\theta$ 是锐角,余弦为正:$\cos\theta = 0.8$。
A trig identity is true for every angle, so it lets you swap an expression for an equivalent one. The master identity is the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$; combined with the reciprocal and quotient identities, it rewrites any trig expression in sine and cosine form.
一个三角恒等式对每一个角都成立,所以它让你把一个表达式换成一个等价的。最核心的是勾股恒等式 $\sin^2\theta + \cos^2\theta = 1$;结合倒数和商恒等式,它能把任何三角表达式改写成正弦和余弦的形式。