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cos t = x cos t = x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
sin t = y sin t = y | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
cos 2 t + sin 2 t = 1 cos 2 t + sin 2 t = 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
tan t = sin t cos t tan t = sin t cos t | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
sec t = 1 cos t sec t = 1 cos t | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
csc t = 1 sin t csc t = 1 sin t | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
cot t = 1 tan t = cos t sin t cot t = 1 tan t = cos t sin t | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
adjacent side : in a right triangle, the side between a given angle and the right angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
angle : the union of two rays having a common endpoint | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
angle of depression : the angle between the horizontal and the line from the object to the observerâs eye, assuming the object is positioned lower than the observer | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
angle of elevation : the angle between the horizontal and the line from the object to the observerâs eye, assuming the object is positioned higher than the observer | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
angular speed : the angle through which a rotating object travels in a unit of time | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
arc length : the length of the curve formed by an arc | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
area of a sector : area of a portion of a circle bordered by two radii and the intercepted arc; the fractionθ2Ï.θ2Ï.multiplied by the area of the entire circle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
cosecant : the reciprocal of the sine function: on the unit circle,csct=1y,yâ0csct=1y,yâ0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
cosine function : thex-value of the point on a unit circle corresponding to a given angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
cotangent : the reciprocal of the tangent function: on the unit circle,cott=xy,yâ0cott=xy,yâ0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
coterminal angles : description of positive and negative angles in standard position sharing the same terminal side | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
degree : a unit of measure describing the size of an angle as one-360th of a full revolution of a circle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
hypotenuse : the side of a right triangle opposite the right angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
identities : statements that are true for all values of the input on which they are defined | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
initial side : the side of an angle from which rotation begins | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
linear speed : the distance along a straight path a rotating object travels in a unit of time; determined by the arc length | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
measure of an angle : the amount of rotation from the initial side to the terminal side | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
negative angle : description of an angle measured clockwise from the positivex-axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
opposite side : in a right triangle, the side most distant from a given angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
period : the smallest intervalPPof a repeating functionffsuch thatf(x+P)=f(x)f(x+P)=f(x) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
positive angle : description of an angle measured counterclockwise from the positivex-axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
Pythagorean Identity : a corollary of the Pythagorean Theorem stating that the square of the cosine of a given angle plus the square of the sine of that angle equals 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
quadrantal angle : an angle whose terminal side lies on an axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
radian : the measure of a central angle of a circle that intercepts an arc equal in length to the radius of that circle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
radian measure : the ratio of the arc length formed by an angle divided by the radius of the circle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
ray : one point on a line and all points extending in one direction from that point; one side of an angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
reference angle : the measure of the acute angle formed by the terminal side of the angle and the horizontal axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
secant : the reciprocal of the cosine function: on the unit circle,sect=1x,xâ0sect=1x,xâ0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
sine function : they-value of the point on a unit circle corresponding to a given angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
standard position : the position of an angle having the vertex at the origin and the initial side along the positivex-axis | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
tangent : the quotient of the sine and cosine: on the unit circle,tant=yx,xâ0tant=yx,xâ0 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
terminal side : the side of an angle at which rotation ends | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
unit circle : a circle with a center at(0,0)(0,0)and radius 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
vertex : the common endpoint of two rays that form an angle | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-terms |
Periodic functions repeat after a given value. The smallest such value is the period. The basic sine and cosine functions have a period of2Ï.2Ï. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The functionsinxsinxis odd, so its graph is symmetric about the origin. The functioncosxcosxis even, so its graph is symmetric about they-axis. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The graph of a sinusoidal function has the same general shape as a sine or cosine function. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
In the general formula for a sinusoidal function, the period isP=2Ï|B|.P=2Ï|B|.SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
In the general formula for a sinusoidal function,|A||A|represents amplitude. If|A|>1,|A|>1,the function is stretched, whereas if|A|<1,|A|<1,the function is compressed. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The valueCBCBin the general formula for a sinusoidal function indicates the phase shift. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The valueDDin the general formula for a sinusoidal function indicates the vertical shift from the midline. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
Combinations of variations of sinusoidal functions can be detected from an equation. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The equation for a sinusoidal function can be determined from a graph. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
A function can be graphed by identifying its amplitude and period. SeeExample 8andExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
A function can also be graphed by identifying its amplitude, period, phase shift, and horizontal shift. SeeExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
Sinusoidal functions can be used to solve real-world problems. SeeExample 11,Example 12, andExample 13. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The tangent function has periodÏ.Ï. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
f(x)=Atan(BxâC)+Df(x)=Atan(BxâC)+Dis a tangent with vertical and/or horizontal stretch/compression and shift. SeeExample 1,Example 2, andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The secant and cosecant are both periodic functions with a period of2Ï.2Ï.f(x)=Asec(BxâC)+Df(x)=Asec(BxâC)+Dgives a shifted, compressed, and/or stretched secant function graph. SeeExample 4andExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
f(x)=Acsc(BxâC)+Df(x)=Acsc(BxâC)+Dgives a shifted, compressed, and/or stretched cosecant function graph. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The cotangent function has periodÏÏand vertical asymptotes at0,±Ï,±2Ï,...0,±Ï,±2Ï,... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The range of cotangent is(ââ,â),(ââ,â),and the function is decreasing at each point in its range. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
The cotangent is zero at±Ï2,±3Ï2,...±Ï2,±3Ï2,... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
f(x)=Acot(BxâC)+Df(x)=Acot(BxâC)+Dis a cotangent with vertical and/or horizontal stretch/compression and shift. SeeExample 8andExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
Real-world scenarios can be solved using graphs of trigonometric functions. SeeExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
An inverse function is one that âundoesâ another function. The domain of an inverse function is the range of the original function and the range of an inverse function is the domain of the original function. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
Because the trigonometric functions are not one-to-one on their natural domains, inverse trigonometric functions are defined for restricted domains. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
For any trigonometric functionf(x),f(x),ifx=fâ1(y),x=fâ1(y),thenf(x)=y.f(x)=y.However,f(x)=yf(x)=yonly impliesx=fâ1(y)x=fâ1(y)ifxxis in the restricted domain off.f.SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
Special angles are the outputs of inverse trigonometric functions for special input values; for example,Ï4=tanâ1(1)andÏ6=sinâ1(12).Ï4=tanâ1(1)andÏ6=sinâ1(12).SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
A calculator will return an angle within the restricted domain of the original trigonometric function. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
Inverse functions allow us to find an angle when given two sides of a right triangle. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
In function composition, if the inside function is an inverse trigonometric function, then there are exact expressions; for example,sin(cosâ1(x))=1âx2.sin(cosâ1(x))=1âx2.SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
If the inside function is a trigonometric function, then the only possible combinations aresinâ1(cosx)=Ï2âxsinâ1(cosx)=Ï2âxif0â¤xâ¤Ï0â¤xâ¤Ïandcosâ1(sinx)=Ï2âxcosâ1(sinx)=Ï2âxifâÏ2â¤xâ¤Ï2.âÏ2â¤xâ¤Ï2.SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
When evaluating the composition of a trigonometric function with an inverse trigonometric function, draw a reference triangle to assist in determining the ratio of sides that represents the output of the trigonometric function. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
When evaluating the composition of a trigonometric function with an inverse trigonometric function, you may use trig identities to assist in determining the ratio of sides. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-concepts |
f ( x ) = A sin ( B x â C ) + D f ( x ) = A cos ( B x â C ) + D f ( x ) = A sin ( B x â C ) + D f ( x ) = A cos ( B x â C ) + D | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
Shifted, compressed, and/or stretched tangent function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
y = A tan ( B x â C ) + D y = A tan ( B x â C ) + D | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
Shifted, compressed, and/or stretched secant function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
y = A sec ( B x â C ) + D y = A sec ( B x â C ) + D | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
Shifted, compressed, and/or stretched cosecant function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
y = A csc ( B x â C ) + D y = A csc ( B x â C ) + D | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
Shifted, compressed, and/or stretched cotangent function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
y = A cot ( B x â C ) + D y = A cot ( B x â C ) + D | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-equations |
amplitude : the vertical height of a function; the constantAAappearing in the definition of a sinusoidal function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
arccosine : another name for the inverse cosine;arccosx=cosâ1xarccosx=cosâ1x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
arcsine : another name for the inverse sine;arcsinx=sinâ1xarcsinx=sinâ1x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
arctangent : another name for the inverse tangent;arctanx=tanâ1xarctanx=tanâ1x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
inverse cosine function : the functioncosâ1x,cosâ1x,which is the inverse of the cosine function and the angle that has a cosine equal to a given number | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
inverse sine function : the functionsinâ1x,sinâ1x,which is the inverse of the sine function and the angle that has a sine equal to a given number | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
inverse tangent function : the functiontanâ1x,tanâ1x,which is the inverse of the tangent function and the angle that has a tangent equal to a given number | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
midline : the horizontal liney=D,y=D,whereDDappears in the general form of a sinusoidal function | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
periodic function : a functionf(x)f(x)that satisfiesf(x+P)=f(x)f(x+P)=f(x)for a specific constantPPand any value ofxx | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
phase shift : the horizontal displacement of the basic sine or cosine function; the constantCBCB | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
sinusoidal function : any function that can be expressed in the formf(x)=Asin(BxâC)+Df(x)=Asin(BxâC)+Dorf(x)=Acos(BxâC)+Df(x)=Acos(BxâC)+D | https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-key-terms |
There are multiple ways to represent a trigonometric expression. Verifying the identities illustrates how expressions can be rewritten to simplify a problem. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
Graphing both sides of an identity will verify it. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
Simplifying one side of the equation to equal the other side is another method for verifying an identity. SeeExample 2andExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
The approach to verifying an identity depends on the nature of the identity. It is often useful to begin on the more complex side of the equation. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
We can create an identity and then verify it. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
Verifying an identity may involve algebra with the fundamental identities. SeeExample 6andExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
Algebraic techniques can be used to simplify trigonometric expressions. We use algebraic techniques throughout this text, as they consist of the fundamental rules of mathematics. SeeExample 8,Example 9, andExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
The sum formula for cosines states that the cosine of the sum of two angles equals the product of the cosines of the angles minus the product of the sines of the angles. The difference formula for cosines states that the cosine of the difference of two angles equals the product of the cosines of the angles plus the pro... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/9-key-concepts |
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