When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in
[link] . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.
Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.
Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.
If needed, draw the right triangle and label the angle provided.
Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
Find the required function:
sine as the ratio of the opposite side to the hypotenuse
cosine as the ratio of the adjacent side to the hypotenuse
tangent as the ratio of the opposite side to the adjacent side
secant as the ratio of the hypotenuse to the adjacent side
cosecant as the ratio of the hypotenuse to the opposite side
cotangent as the ratio of the adjacent side to the opposite side
Evaluating trigonometric functions of angles not in standard position
Finding trigonometric functions of special angles using side lengths
It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples of
and
Remember, however, that when dealing with right triangles, we are limited to angles between
Suppose we have a
triangle, which can also be described as a
triangle. The sides have lengths in the relation
The sides of a
triangle, which can also be described as a
triangle, have lengths in the relation
These relations are shown in
[link] .
We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.
Given trigonometric functions of a special angle, evaluate using side lengths.
Use the side lengths shown in
[link] for the special angle you wish to evaluate.
Use the ratio of side lengths appropriate to the function you wish to evaluate.
Evaluating trigonometric functions of special angles using side lengths
Find the exact value of the trigonometric functions of
using side lengths.
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?