The tangent of an angle is the ratio of the
y -value to the
x -value of the corresponding point on the unit circle.
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.
The six trigonometric functions can be found from a point on the unit circle. See
[link] .
Trigonometric functions can also be found from an angle. See
[link] .
Trigonometric functions of angles outside the first quadrant can be determined using reference angles. See
[link] .
A function is said to be even if
and odd if
Cosine and secant are even; sine, tangent, cosecant, and cotangent are odd.
Even and odd properties can be used to evaluate trigonometric functions. See
[link] .
The Pythagorean Identity makes it possible to find a cosine from a sine or a sine from a cosine.
Identities can be used to evaluate trigonometric functions. See
[link] and
[link] .
Fundamental identities such as the Pythagorean Identity can be manipulated algebraically to produce new identities. See
[link] .
The trigonometric functions repeat at regular intervals.
The period
of a repeating function
is the smallest interval such that
for any value of
The values of trigonometric functions of special angles can be found by mathematical analysis.
To evaluate trigonometric functions of other angles, we can use a calculator or computer software. See
[link] .
Section exercises
Verbal
On an interval of
can the sine and cosine values of a radian measure ever be equal? If so, where?
Yes, when the reference angle is
and the terminal side of the angle is in quadrants I and III. Thus, at
the sine and cosine values are equal.
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
from theory: distance [miles] = speed [mph] × time [hours]
info #1
speed_Dennis × 1.5 = speed_Wayne × 2
=> speed_Wayne = 0.75 × speed_Dennis (i)
info #2
speed_Dennis = speed_Wayne + 7 [mph] (ii)
use (i) in (ii) => [...]
speed_Dennis = 28 mph
speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5.
Substituting the first equation into the second:
W * 2 = (W + 7) * 1.5
W * 2 = W * 1.5 + 7 * 1.5
0.5 * W = 7 * 1.5
W = 7 * 3 or 21
W is 21
D = W + 7
D = 21 + 7
D = 28
Salma
Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67.
Check:
Sales = 3542
Commission 12%=425.04
Pay = 500 + 425.04 = 925.04.
925.04 > 925.00
Munster
difference between rational and irrational numbers
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?