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We use the command âLnRegâ on a graphing utility to fit a function of the formy=a+bln(x)y=a+bln(x)to a set of data points. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-concepts |
Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows as the function approaches an upper limit. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-concepts |
We use the command âLogisticâ on a graphing utility to fit a function of the formy=c1+aeâbxy=c1+aeâbxto a set of data points. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-concepts |
f ( x ) = b x , where b > 0 , b â 1 f ( x ) = b x , where b > 0 , b â 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
f ( x ) = a b x , where a > 0 , b > 0 , b â 1 f ( x ) = a b x , where a > 0 , b > 0 , b â 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
A ( t ) = P ( 1 + r n ) n t , where A ( t ) is the account value at time t t is the number of years P is the initial investment, often called the principal r is the annual percentage rate (APR), or nominal rate n is the number of compounding periods in one year A ( t ) = P ( 1 + r n ) n t , where A ( t ) is the account... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
A ( t ) = a e r t , where A ( t ) = a e r t , where t t is the number of unit time periods of growth a a is the starting amount (in the continuous compounding formula a is replaced with P, the principal) e e is the mathematical constant, e â 2.718282 e â 2.718282 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
General Form for the Translation of the Parent Function f ( x ) = b x f ( x ) = b x | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
f ( x ) = a b x + c + d f ( x ) = a b x + c + d | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
For x > 0 , b > 0 , b â 1 , x > 0 , b > 0 , b â 1 , y = log b ( x ) y = log b ( x ) if and only if b y = x . b y = x . | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
For x > 0 , x > 0 , y = log ( x ) y = log ( x ) if and only if 10 y = x . 10 y = x . | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
For x > 0 , x > 0 , y = ln ( x ) y = ln ( x ) if and only if e y = x . e y = x . | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
General Form for the Translation of the Parent Logarithmic Function f ( x ) = log b ( x ) f ( x ) = log b ( x ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
f ( x ) = a log b ( x + c ) + d f ( x ) = a log b ( x + c ) + d | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
log b ( M N ) = log b ( M ) + log b ( N ) log b ( M N ) = log b ( M ) + log b ( N ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
log b ( M N ) = log b M â log b N log b ( M N ) = log b M â log b N | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
log b ( M n ) = n log b M log b ( M n ) = n log b M | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
log b M = log n M log n b n > 0 , n â 1 , b â 1 log b M = log n M log n b n > 0 , n â 1 , b â 1 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
For any algebraic expressions S S and T T and any positive real number b , b , where b S = b T b S = b T if and only if S = T . S = T . | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
For any algebraic expression S and positive real numbers b b and c , c , where b â 1 , b â 1 , log b ( S ) = c log b ( S ) = c if and only if b c = S . b c = S . | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
For any algebraic expressions S and T and any positive real number b , b , where b â 1 , b â 1 , log b S = log b T log b S = log b T if and only if S = T . S = T . | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
If A = A 0 e k t , A = A 0 e k t , k < 0 , k < 0 , the half-life is t = â ln ( 2 ) k . t = â ln ( 2 ) k . | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
t = ln ( A A 0 ) â 0.000121 . t = ln ( A A 0 ) â 0.000121 . A 0 A 0 is the amount of carbon-14 when the plant or animal died A A is the amount of carbon-14 remaining today t t is the age of the fossil in years | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
If A = A 0 e k t , A = A 0 e k t , k > 0 , k > 0 , the doubling time is t = ln 2 k t = ln 2 k | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
T ( t ) = A e k t + T s , T ( t ) = A e k t + T s , where T s T s is the ambient temperature, A = T ( 0 ) â T s , A = T ( 0 ) â T s , and k k is the continuous rate of cooling. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-equations |
annual percentage rate (APR) : the yearly interest rate earned by an investment account, also callednominal rate | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
carrying capacity : in a logistic model, the limiting value of the output | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
change-of-base formula : a formula for converting a logarithm with any base to a quotient of logarithms with any other base. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
common logarithm : the exponent to which 10 must be raised to getx;x;log10(x)log10(x)is written simply aslog(x).log(x). | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
compound interest : interest earned on the total balance, not just the principal | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
doubling time : the time it takes for a quantity to double | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
exponential growth : a model that grows by a rate proportional to the amount present | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
extraneous solution : a solution introduced while solving an equation that does not satisfy the conditions of the original equation | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
half-life : the length of time it takes for a substance to exponentially decay to half of its original quantity | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
logarithm : the exponent to whichbbmust be raised to getx;x;writteny=logb(x)y=logb(x) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
logistic growth model : a function of the formf(x)=c1+aeâbxf(x)=c1+aeâbxwherec1+ac1+ais the initial value,ccis the carrying capacity, or limiting value, andbbis a constant determined by the rate of growth | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
natural logarithm : the exponent to which the numbereemust be raised to getx;x;loge(x)loge(x)is written asln(x).ln(x). | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
Newtonâs Law of Cooling : the scientific formula for temperature as a function of time as an objectâs temperature is equalized with the ambient temperature | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
nominal rate : the yearly interest rate earned by an investment account, also calledannual percentage rate | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
order of magnitude : the power of ten, when a number is expressed in scientific notation, with one non-zero digit to the left of the decimal | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
power rule for logarithms : a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
product rule for logarithms : a rule of logarithms that states that the log of a product is equal to a sum of logarithms | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
quotient rule for logarithms : a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms | https://openstax.org/books/algebra-and-trigonometry-2e/pages/6-key-terms |
An angle is formed from the union of two rays, by keeping the initial side fixed and rotating the terminal side. The amount of rotation determines the measure of the angle. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
An angle is in standard position if its vertex is at the origin and its initial side lies along the positivex-axis. A positive angle is measured counterclockwise from the initial side and a negative angle is measured clockwise. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
To draw an angle in standard position, draw the initial side along the positivex-axis and then place the terminal side according to the fraction of a full rotation the angle represents. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
In addition to degrees, the measure of an angle can be described in radians. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
To convert between degrees and radians, use the proportionθ180=θRÏ.θ180=θRÏ.SeeExample 3andExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Two angles that have the same terminal side are called coterminal angles. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
We can find coterminal angles by adding or subtracting360°360°or2Ï.2Ï.SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Coterminal angles can be found using radians just as they are for degrees. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The length of a circular arc is a fraction of the circumference of the entire circle. SeeExample 8. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The area of sector is a fraction of the area of the entire circle. SeeExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
An object moving in a circular path has both linear and angular speed. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The angular speed of an object traveling in a circular path is the measure of the angle through which it turns in a unit of time. SeeExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The linear speed of an object traveling along a circular path is the distance it travels in a unit of time. SeeExample 11. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
We can define trigonometric functions as ratios of the side lengths of a right triangle. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The same side lengths can be used to evaluate the trigonometric functions of either acute angle in a right triangle. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
We can evaluate the trigonometric functions of special angles, knowing the side lengths of the triangles in which they occur. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Any two complementary angles could be the two acute angles of a right triangle. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
If two angles are complementary, the cofunction identities state that the sine of one equals the cosine of the other and vice versa. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
We can use trigonometric functions of an angle to find unknown side lengths. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Select the trigonometric function representing the ratio of the unknown side to the known side. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Right-triangle trigonometry facilitates the measurement of inaccessible heights and distances. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The unknown height or distance can be found by creating a right triangle in which the unknown height or distance is one of the sides, and another side and angle are known. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Finding the function values for the sine and cosine begins with drawing a unit circle, which is centered at the origin and has a radius of 1 unit. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Using the unit circle, the sine of an anglettequals they-value of the endpoint on the unit circle of an arc of lengthttwhereas the cosine of an anglettequals thex-value of the endpoint. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The sine and cosine values are most directly determined when the corresponding point on the unit circle falls on an axis. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
When the sine or cosine is known, we can use the Pythagorean Identity to find the other. The Pythagorean Identity is also useful for determining the sines and cosines of special angles. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Calculators and graphing software are helpful for finding sines and cosines if the proper procedure for entering information is known. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The domain of the sine and cosine functions is all real numbers. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The range of both the sine and cosine functions is[â1,1].[â1,1]. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The sine and cosine of an angle have the same absolute value as the sine and cosine of its reference angle. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The signs of the sine and cosine are determined from thex- andy-values in the quadrant of the original angle. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
An angleâs reference angle is the size angle,t,t,formed by the terminal side of the anglettand the horizontal axis. SeeExample 5. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Reference angles can be used to find the sine and cosine of the original angle. SeeExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Reference angles can also be used to find the coordinates of a point on a circle. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The tangent of an angle is the ratio of they-value to thex-value of the corresponding point on the unit circle. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The secant, cotangent, and cosecant are all reciprocals of other functions. The secant is the reciprocal of the cosine function, the cotangent is the reciprocal of the tangent function, and the cosecant is the reciprocal of the sine function. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The six trigonometric functions can be found from a point on the unit circle. SeeExample 1. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Trigonometric functions can also be found from an angle. SeeExample 2. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Trigonometric functions of angles outside the first quadrant can be determined using reference angles. SeeExample 3. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
A function is said to be even iff(âx)=f(x)f(âx)=f(x)and odd iff(âx)=âf(x)f(âx)=âf(x)for allxin the domain off. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Even and odd properties can be used to evaluate trigonometric functions. SeeExample 4. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Identities can be used to evaluate trigonometric functions. SeeExample 5andExample 6. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. SeeExample 7. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The trigonometric functions repeat at regular intervals. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The periodPPof a repeating functionffis the smallest interval such thatf(x+P)=f(x)f(x+P)=f(x)for any value ofx.x. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
The values of trigonometric functions can be found by mathematical analysis. SeeExample 8andExample 9. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
To evaluate trigonometric functions of other angles, we can use a calculator or computer software. SeeExample 10. | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-concepts |
s = r θ s = r θ | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
A = 1 2 θ r 2 A = 1 2 θ r 2 | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
Ï = θ t Ï = θ t | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
v = s t v = s t | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
v = r Ï v = r Ï | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
Sine sin t = opposite hypotenuse Cosine cos t = adjacent hypotenuse Tangent tan t = opposite adjacent Secant sec t = hypotenuse adjacent Cosecant csc t = hypotenuse opposite Cotangent cot t = adjacent opposite Sine sin t = opposite hypotenuse Cosine cos t = adjacent hypotenuse Tangent tan t = opposite adjacent Secant s... | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
sin t = 1 csc t csc t = 1 sin t cos t = 1 sec t sec t = 1 cos t tan t = 1 cot t cot t = 1 tan t sin t = 1 csc t csc t = 1 sin t cos t = 1 sec t sec t = 1 cos t tan t = 1 cot t cot t = 1 tan t | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
cos t = sin ( Ï 2 â t ) sin t = cos ( Ï 2 â t ) tan t = cot ( Ï 2 â t ) cot t = tan ( Ï 2 â t ) sec t = csc ( Ï 2 â t ) cos t = sin ( Ï 2 â t ) sin t = cos ( Ï 2 â t ) tan t = cot ( Ï 2 â t ) cot t = tan ( Ï 2 â t ) sec t = csc ( Ï 2 â t ) | https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-key-equations |
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