06 Trigonometric Identities and Equations Free Online FlashCards

How does an identity differ from a trigonometric equation?

An identity is true for every angle in the common domain, whereas a trigonometric equation is true only for particular values of the variable.

State the reciprocal identities.

$\csc\theta=\frac{1}{\sin\theta}$, $\sec\theta=\frac{1}{\cos\theta}$, and $\cot\theta=\frac{1}{\tan\theta}$.

State the quotient identities for tangent and cotangent.

$\tan\theta=\frac{\sin\theta}{\cos\theta}$ and $\cot\theta=\frac{\cos\theta}{\sin\theta}$.

What identity results from dividing $\sin^2\theta+\cos^2\theta=1$ by $\cos^2\theta$?

$1+\tan^2\theta=\sec^2\theta$. It follows by dividing $\sin^2\theta+\cos^2\theta=1$ by $\cos^2\theta$.

Which fundamental trigonometric functions are even?

$\cos(-\theta)=\cos\theta$ and $\sec(-\theta)=\sec\theta$. Thus, cosine and secant are even functions.

What is the standard method for verifying a trigonometric identity?

Transform one side into the other using valid algebraic steps, usually starting with the more complicated side. Preserve the common domain by excluding zero denominators.

What is the cosine difference formula?

$\cos(\alpha-\beta)=\cos\alpha\cos\beta+\sin\alpha\sin\beta$. A difference inside the cosine produces addition between the products.

State the tangent sum formula.

$\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}$, where the expression is defined.

Which double-angle identity expresses $\cos(2\theta)$ using only sine?

$\cos(2\theta)=1-2\sin^2\theta$. This form is useful when an equation already contains powers of sine.

What determines the sign in a half-angle formula?

The sign is determined by the quadrant containing $\frac{\alpha}{2}$, not necessarily by the quadrant containing $\alpha$.

Find the exact value of $\sin\frac{\pi}{8}$.

$\sin\frac{\pi}{8}=\frac{\sqrt{2-\sqrt2}}{2}$. The positive root is used because $\frac{\pi}{8}$ lies in Quadrant I.

What is the general solution of $\tan x=a$?

The general solution is $x=\arctan(a)+\pi k$, where $k\in\mathbb Z$. Tangent repeats every $\pi$.