Trigonometric Identities Flashcards

All 6 cards in this deck

State the identity linking $\tan\theta$ with $\sin\theta$ and $\cos\theta$.

$\tan\theta = \dfrac{\sin\theta}{\cos\theta}$

State the identity connecting $\sin^2\theta$ and $\cos^2\theta$.

$\sin^2\theta + \cos^2\theta = 1$

True or false? $\sin^2\theta$ means $\sin(\theta^2)$.

False. $\sin^2\theta$ means $(\sin\theta)^2$ — square the value of $\sin\theta$.

How do you find $\cos\theta$ exactly when you are given $\sin\theta$ and $\theta$ is acute?
e.g. $\sin\theta = \frac{3}{5}$

1. Use $\sin^2\theta + \cos^2\theta = 1$: $\cos^2\theta = 1 - \left(\frac{3}{5}\right)^2 = \frac{16}{25}$
2. Square root, choosing the sign for the quadrant: $\cos\theta = \frac{4}{5}$

What are the steps to prove a trigonometric identity?
e.g. prove $\tan\theta\cos\theta \equiv \sin\theta$

1. Start with the more complicated side: LHS $= \tan\theta\cos\theta$
2. Replace $\tan\theta$ by $\frac{\sin\theta}{\cos\theta}$: $\frac{\sin\theta}{\cos\theta}\times\cos\theta$
3. Simplify to reach the other side: $=\sin\theta=$ RHS

How do you solve an equation of the form $a\sin x = b\cos x$?
e.g. $\sin x = \cos x$ for $0° \le x \le 360°$

1. Divide both sides by $\cos x$ to get a single ratio: $\tan x = 1$
2. Solve, adding $180°$ for the second solution: $x = 45°$ and $x = 225°$