Trigonometry

78 flashcards by MentisLibrary

From SOHCAHTOA to the unit circle, identities, and the laws of sines and cosines. Exact values, graph transformations, and the sign rules that make every angle problem quick.

What does SOHCAHTOA stand for?

$\sin = \dfrac{\text{opposite}}{\text{hypotenuse}}$, $\cos = \dfrac{\text{adjacent}}{\text{hypotenuse}}$, $\tan = \dfrac{\text{opposite}}{\text{adjacent}}$

What are the three reciprocal trigonometric functions?

$\csc\theta = \dfrac{1}{\sin\theta}$, $\sec\theta = \dfrac{1}{\cos\theta}$, $\cot\theta = \dfrac{1}{\tan\theta}$

What is the quotient identity for tangent?

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

What is the unit circle?

The circle of radius $1$ centered at the origin, where a point at angle $\theta$ has coordinates $(\cos\theta, \sin\theta)$

How do you convert between degrees and radians?

Multiply degrees by $\dfrac{\pi}{180}$, or radians by $\dfrac{180}{\pi}$

What is a radian?

The angle subtending an arc equal in length to the radius

What is the arc length formula in radians?

$s = r\theta$ with $\theta$ in radians

What are $\sin 30°$, $\cos 30°$, and $\tan 30°$?

$\sin 30° = \dfrac{1}{2}$, $\cos 30° = \dfrac{\sqrt{3}}{2}$, $\tan 30° = \dfrac{\sqrt{3}}{3}$

What are $\sin 45°$, $\cos 45°$, and $\tan 45°$?

$\sin 45° = \cos 45° = \dfrac{\sqrt{2}}{2}$ and $\tan 45° = 1$

What are $\sin 60°$, $\cos 60°$, and $\tan 60°$?

$\sin 60° = \dfrac{\sqrt{3}}{2}$, $\cos 60° = \dfrac{1}{2}$, $\tan 60° = \sqrt{3}$

What are the sine and cosine of $0$, $\frac{\pi}{2}$, $\pi$, and $\frac{3\pi}{2}$?

$\sin$: $0, 1, 0, -1$. $\cos$: $1, 0, -1, 0$

What is a reference angle?

The acute angle between the terminal side and the $x$-axis

What does ASTC tell you about signs by quadrant?

Quadrant I: All positive. II: Sine positive. III: Tangent positive. IV: Cosine positive

What are coterminal angles?

Angles sharing the same terminal side, differing by whole rotations of $360°$ or $2\pi$

What is the fundamental Pythagorean identity?

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

What are the other two Pythagorean identities?

$1 + \tan^2\theta = \sec^2\theta$ and $1 + \cot^2\theta = \csc^2\theta$

What are the cofunction identities?

$\sin\theta = \cos(90° - \theta)$, $\tan\theta = \cot(90° - \theta)$, $\sec\theta = \csc(90° - \theta)$

Which trig functions are even and which are odd?

Cosine and secant are even, so $\cos(-\theta) = \cos\theta$. Sine, tangent, cosecant, and cotangent are odd

What is the sine sum formula?

$\sin(A+B) = \sin A \cos B + \cos A \sin B$

What is the cosine sum formula?

$\cos(A+B) = \cos A \cos B - \sin A \sin B$

What is the tangent sum formula?

$\tan(A+B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$

What is the sine double angle formula?

$\sin 2\theta = 2\sin\theta\cos\theta$

What are the three forms of the cosine double angle formula?

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

What is the tangent double angle formula?

$\tan 2\theta = \dfrac{2\tan\theta}{1 - \tan^2\theta}$

What are the half angle formulas for sine and cosine?

$\sin\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1-\cos\theta}{2}}$ and $\cos\dfrac{\theta}{2} = \pm\sqrt{\dfrac{1+\cos\theta}{2}}$

What are the power reducing identities?

$\sin^2\theta = \dfrac{1-\cos 2\theta}{2}$ and $\cos^2\theta = \dfrac{1+\cos 2\theta}{2}$

What is the law of sines?

$\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}$

What is the ambiguous case of the law of sines?

With SSA, the given data may produce no triangle, one triangle, or two

What is the law of cosines?

$c^2 = a^2 + b^2 - 2ab\cos C$

How do you find an angle when you know all three sides?

Rearrange the law of cosines: $\cos C = \dfrac{a^2+b^2-c^2}{2ab}$