The next set of identities is the set of
half-angle formulas , which can be derived from the reduction formulas and we can use when we have an angle that is half the size of a special angle. If we replace
with
the half-angle formula for sine is found by simplifying the equation and solving for
Note that the half-angle formulas are preceded by a
sign. This does not mean that both the positive and negative expressions are valid. Rather, it depends on the quadrant in which
terminates.
The half-angle formula for sine is derived as follows:
To derive the half-angle formula for cosine, we have
For the tangent identity, we have
Half-angle formulas
The
half-angle formulas are as follows:
Using a half-angle formula to find the exact value of a sine function
Given the tangent of an angle and the quadrant in which the angle lies, find the exact values of trigonometric functions of half of the angle.
Draw a triangle to represent the given information.
Determine the correct half-angle formula.
Substitute values into the formula based on the triangle.
Simplify.
Finding exact values using half-angle identities
Given that
and
lies in quadrant III, find the exact value of the following:
Using the given information, we can draw the triangle shown in
[link] . Using the Pythagorean Theorem, we find the hypotenuse to be 17. Therefore, we can calculate
and
Before we start, we must remember that, if
is in quadrant III, then
so
This means that the terminal side of
is in quadrant II, since
To find
we begin by writing the half-angle formula for sine. Then we substitute the value of the cosine we found from the triangle in
[link] and simplify.
We choose the positive value of
because the angle terminates in quadrant II and sine is positive in quadrant II.
To find
we will write the half-angle formula for cosine, substitute the value of the cosine we found from the triangle in
[link] , and simplify.
We choose the negative value of
because the angle is in quadrant II because cosine is negative in quadrant II.
To find
we write the half-angle formula for tangent. Again, we substitute the value of the cosine we found from the triangle in
[link] and simplify.
We choose the negative value of
because
lies in quadrant II, and tangent is negative in quadrant II.
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?