Equivalent Trigonometric Expressions
Introduced| English |
|---|
| identity/aɪˈdentɪti/ |
| equivalent/ɪˈkwɪvələnt/ |
| Pythagorean identity/ˈpaɪθəɡɔːrɪən aɪˈdentɪti/ |
| sine/saɪn/ |
| cosine/ˈkəʊsaɪn/ |
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.
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.
Watch the Pythagorean identity hold
For any angle, the point on the unit circle satisfies cos squared plus sin squared equals 1 — the master trig identity.
A trigonometric identity is an equation that is true…
An identity holds for all valid inputs — unlike an equation you solve for particular values.
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.
The Pythagorean identity states that…
Because a unit-circle point $(\cos\theta, \sin\theta)$ has radius $1$: $\cos^2\theta + \sin^2\theta = 1$.
Select all true statements about trig identities.
An identity is true for all angles, not just one. The other three are correct.
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.
If $\sin\theta = 0.6$, then from the Pythagorean identity $\cos^2\theta = 1 - 0.36 =$ ____.
$\cos^2\theta = 1 - \sin^2\theta = 1 - 0.36 = 0.64$, so $\cos\theta = \pm 0.8$.
Identities let you rewrite a trig expression as an equivalent one that may be easier to work with.
That is their main use: swap in an equal form to simplify an expression or solve an equation.
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.
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.
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: $\cos\theta = 0.8$.
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.