Flashcards: Trigonometric Functions — 58 cards

What is an angle in geometry?

The measure of rotation of a ray about its initial point.

What is the original ray in an angle called?

The initial side.

What is the rotated ray in an angle called?

The terminal side.

What is the rotation point of an angle called?

The vertex.

What kind of rotation produces a positive angle?

An anticlockwise rotation.

What kind of rotation produces a negative angle?

A clockwise rotation.

How much of a revolution is one degree?

One three-hundred-and-sixtieth of a revolution.

How many minutes are in one degree?

Sixty minutes.

What is one radian in a unit circle?

An angle subtended by an arc of length one unit at the centre.

What is the relation between degrees and radians?

$180^\circ = \pi \text{ radians}$.

How do you convert degrees to radians?

Multiply degrees by $\frac{\pi}{180}$.

How do you convert radians to degrees?

Multiply radians by $\frac{180}{\pi}$.

What is the radian equivalent of 30 degrees?

$\frac{\pi}{6}$ radians.

What is the radian equivalent of 45 degrees?

$\frac{\pi}{4}$ radians.

What is the radian equivalent of 90 degrees?

$\frac{\pi}{2}$ radians.

What is the radian equivalent of 360 degrees?

$2\pi$ radians.

What is the formula relating arc length l, radius r, and angle θ in radians?

$l = r\theta$

How can the central angle θ be found from arc length l and radius r?

$\theta = \frac{l}{r}$

How is the distance travelled by a rotating hand's tip found?

By expressing rotation in radians and applying the arc-length relation.

What ratio do the radii have if arcs of equal length subtend 65° and 110°?

The radii are in the ratio 22:13.

For point P(a,b) on unit circle at angle x, what is cos x?

Cos x equals a, the x-coordinate of P.

For point P(a,b) on unit circle at angle x, what is sin x?

Sin x equals b, the y-coordinate of P.

What is the fundamental identity relating sin x and cos x?

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

What defines a quadrantal angle?

It is an integral multiple of $\frac{\pi}{2}$.

Name some examples of quadrantal angles.

Examples include 0, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$.

What is sin 0, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$?

They are 0, 1, 0, −1, and 0 respectively.

What is cos 0, $\frac{\pi}{2}$, $\pi$, $\frac{3\pi}{2}$, and $2\pi$?

They are 1, 0, −1, 0, and 1 respectively.

What is the formula for cosec x in terms of sine?

Cosec x equals 1 divided by sin x.

In which quadrant are sine and cosine both positive?

Quadrant I.

What identity relates $1+\tan^2x$ to a reciprocal function?

$1+\tan^2x=\sec^2x$.

What are the range limits for sine for all real x?

Sine values lie between -1 and 1.

What is the formula for sec x in terms of cosine?

Sec x equals 1 divided by cos x.

In which quadrant is sine positive but cosine is not?

Quadrant II.

What identity relates $1+\cot^2x$ to a reciprocal function?

$1+\cot^2x=\cosec^2x$.

What are the range limits for cosine for all real x?

Cosine values lie between -1 and 1.

For which x values are sine and cosine undefined?

Sine and cosine are defined for every real number.

Which x values are excluded for cosecant and cotangent?

x = nπ, where n is an integer.

Which x values are excluded for secant and tangent?

x = (2n+1)π/2, where n is an integer.

What is the range of sine and cosine functions?

The range is [−1,1].

What is the range of tangent and cotangent functions?

Their range is all real numbers.

What is the range of secant and cosecant functions?

Their range is y ≤ −1 or y ≥ 1.

After what interval do sine, cosine, secant, and cosecant repeat?

They repeat after 2π.

After what interval do tangent and cotangent repeat?

They repeat after π.

What is the formula for $\sin(x+y)$ in sum identities?

$\sin(x+y)=\sin x\cos y + \cos x\sin y$.

What is the formula for $\sin(x-y)$ in difference identities?

$\sin(x-y)=\sin x\cos y - \cos x\sin y$.

What is the formula for $\cos(x+y)$ in sum identities?

$\cos(x+y)=\cos x\cos y - \sin x\sin y$.

What is the formula for $\cos(x-y)$ in difference identities?

$\cos(x-y)=\cos x\cos y + \sin x\sin y$.

What is the formula for $\tan(x+y)$ when defined?

$\tan(x+y)=\frac{\tan x + \tan y}{1 - \tan x \tan y}$.

What is the complementary-angle identity for $\cos(\pi/2 - x)$?

$\cos(\pi/2 - x) = \sin x$.

What is the complementary-angle identity for $\sin(\pi/2 - x)$?

$\sin(\pi/2 - x) = \cos x$.

What is the double-angle identity for sine?

$\sin2x=2\sin x\cos x$

What is one form of the double-angle identity for cosine?

$\cos2x=\cos^2x-\sin^2x$

What is the tangent double-angle identity?

$\tan2x=\frac{2\tan x}{1-\tan^2x}$ when defined

What is the triple-angle identity for sine?

$\sin3x=3\sin x-4\sin^3x$

What is the triple-angle identity for cosine?

$\cos3x=4\cos^3x-3\cos x$

What is the product-to-sum identity for $2\cos x\cos y$?

$2\cos x\cos y=\cos(x+y)+\cos(x-y)$

What is the sum-to-product identity for $\cos x + \cos y$?

$\cos x+\cos y=2\cos\frac{x+y}{2}\cos\frac{x-y}{2}$

What is the sum-to-product identity for $\sin x + \sin y$?

$\sin x+\sin y=2\sin\frac{x+y}{2}\cos\frac{x-y}{2}$