In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions[1][2]) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena, through Fourier analysis.
Basis of trigonometry: if two right triangles have equal acute angles, they are similar, so their side lengths are proportional. Proportionality constants are written within the image: sin θ, cos θ, tan θ, where θ is the common measure of five acute angles.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.[3]
The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extending these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) is often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane (from which some isolated points are removed).
Contents
Right-angled triangle definitions Edit
A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.
Plot of the six trigonometric functions, the unit circle, and a line for the angle θ = 0.7 radians. The points labelled 1, Sec(θ), Csc(θ) represent the length of the line segment from the origin to that point. Sin(θ), Tan(θ), and 1 are the heights to the line starting from the x-axis, while Cos(θ), 1, and Cot(θ) are lengths along the x-axis starting from the origin.
In this section, the same upper-case letter denotes a vertex of a triangle and the measure of the corresponding angle; the same lower case letter denotes an edge of the triangle and its length.
Given an acute angle A = θ of a right-angled triangle, the hypotenuse h is the side that connects the two acute angles. The side b adjacent to θ is the side of the triangle that connects θ to the right angle. The third side a is said to be opposite to θ.
If the angle θ is given, then all sides of the right-angled triangle are well-defined up to a scaling factor. This means that the ratio of any two side lengths depends only on θ. Thus these six ratios define six functions of θ, which are the trigonometric functions. More precisely, the six trigonometric functions are:[4][5]
sine
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if triangle GHI is equilateral and JHK is isosceles, what is the measure of JHK
Answer:
The measure of JHK is 25 degrees
Step-by-step explanation:
If GHI is equilateral, it means each of the angles are equal to 60 degrees
Thus;
7x - 31 = 60
7x = 60 + 31
7x = 91
x = 91/7
x = 13 degrees
The measure of JHK IS x + 12
simply insert the value of x here
Angle JHK = 13 + 12
Angle JHK is 25 degrees
Answer: 25
Step-by-step explanation:
This table represents a linear function. Find the missing values to complete the table.
Answer:
(2,5) and (7,12.5)
Step-by-step explanation:
Solve for change in y over change in x using the points (3,13/2) and (6,11). You can then get the equation y=1.5x+2. Then plug in the values you are given from the missing points using the equation. Hope this helped! If it did, please mark the brainliest.
Thomas has a bag with 6 purple and 5 blue marbles. If his friend Nick pulls 2 marbles from the bag, what is the probability of him getting 2 purple marbles. 30/121 30/121 21/22 21/22 3/11 3/11 11/21
Answer: 3/11
Step-by-step explanation:
Number of purple marbles = 6
Number of blue marbles = 5
Total number of marbles = 6 + 5 = 11
The probability of picking the dust purple marble will be 6/11. After a purple marvel has been picked, there'll be 5 purple marbles left and the total number of marbles will be 10, therefore the probability of picking the second purple marble will be 5/10.
Therefore we then multiply the probability of picking the first purple marble by the probability of picking the second purple marble. This will be:
= 6/11 × 5/10
= 6/11 × 1/2
= 3/11
ILL MARK BRAINLIEST IF FAST
Select the correct answer.
Which table represents a linear function that has a greater unit rate than that of the function y = 200x + 50?
Answer:
D
Step-by-step explanation:
A unit rate is basically which one increases more as time goes on. Now. to make it easier let me explain visual.
You have 50 apples at the start and every hour you get 200 more apples, sweet!
Now let's look at the chart.
The first one is 200x+100, this is wrong since this has a better start but you don't get as much apples.
Same goes for the B and C. The only one that would make sense is D since you're also adding 10 more apples for each one you increase.
HELP ASAP PLEASE !!!><
WILL GIVE BRAINLIEST ❤️
Find the slope of the line passing through the given points using the slope formula. Describe the slope as
positive, negative, zero, or undefined.
(-5, -18) and (-5, -3)
Answer:
Your slope would be undefined
-[tex]\frac{-3-(-18)}{-5-(-5\\} \\[/tex]
the -5-(-5) is cancelling it's self out
And i believe M=0.75 take this with a bit of salt
hellppppp please simplify the expression- 96
Answer:
Step-by-step explanation:
-√96 = -4√6
jackson usually makes $250 per week working at a grocery store. Last week he made 60% more than usual. This week Jackson hopes to make 7/8 of what he made last week. how much money does jackson hope to make this week?
Answer: 350
250×.60=150
250+150= 400
7÷8= 0.875
0.875×400=350
I hope this is good enough:
WHO LIKES LARRAY
Cause I do
3 +5.2.c = 1 - 2.87
Answer:
-0.0936538462
Step-by-step explanation:
Start by solving the right side of the equation
1 - 2.87 = -1.87
now we have:
3 + 5.2c = -1.87
subtract 3 from both sides to get:
5.2c = -4.87
divide both sides by 5.2 to get:
-0.0936538462
hope that's what you're looking for! :)
The cheerleaders at City High make their own
megaphones by cutting off the small end of a
cone made from heavy paper. If the small end of
the megaphone is to have a radius of 2.5 cm,
what should be the height of the cone that is cut
off?
2.5 cm
60 cm
Answer:
The height of the cone that is cut would be 2.5
' the length of one side of an equilateral is 3x+2. its perimeter is 60 centimeters. write an equation that could be used to find the value of x. solve for x and then find the length of the side. ''
I need to find the equation and the side length :')
also please answer this quickly hhhhhhhhhhhhh
Answer:
Step-by-step explanation:
Triangle equilateral has 3 sides
3(3x+2) = 60
9x+6 =60
9x= 54
x=9
3x9 +2 = 20 One side = 20
20x3=60 3 sides = 60
Answer:
60
Step-by-step explanation: this is simple because it is lou
If x=5 and the area of the rectangle is 30 what is the other length of the rectangle?
Answer:
The other length would be 6.
Step-by-step explanation:
Rectangle area is l times height. So x is either the length for height. So we have 5 times x = 30. To find the other length we have to take 30/5 which equals 6
Factor completely: 125 x cubed space plus space 343
Answer:
(5x+7)(25x^2-35x+49)
Step-by-step explanation:
Not really an explanation for factoring, but I'm sure it's right.
Calculate 8/9-2/3 giving your answer in its simplest form.
Answer:
(8/9) - (2/3) = (8-6)/9 = 2/9 = 0.22
See attachment for math work and answer.
two common multiples of 4 and 8 are
Answer:
8, 16
Step-by-step explanation:
kate ran 4 miles how far did she run in kilometers
Answer:
6.4
Step-by-step explanation:
Answer:
6.5 kilometers
Step-by-step explanation:
Please help me i will give brainlist please...
Answer: All i know is you have to multiply all of those numbers because its area
Step-by-step explanation:
first to answer correctly gets brainleist! (bringing out a lot of questions so yea!)
Answer:
C.
Step-by-step explanation:
The easiest way to tell if a relation is a function is to look at the values of the x's. If more than one x is the same, it cannot be a function. In contrast, if more than one y is the same, it is okay. The answers A, B, and D are functions because the x's are all different. Looking at the table for the answer C, you can see that two x's are the same (there are two x's with the value of 1).
Hope this helps!! Have a wonderful day :D
Sup nerds. Do this for me or I'm gonna get screamed at by my mom also show all the work
Answer:
the answer is c they bother are the same and have the same equals
A train leaves LA at 2:00 pm and heads north at 50 mph. If the next train leaves the station three hours later and heads north at 60 mph, at what time will the second train catch up to the first train?
Answer:
https://brainly.com/question/6423861
Step-by-step explanation:
Use link and give me 5 stars and brainliest
Write an equation in Slope Intercept form with a slope of -5 and y-intercept of 4. (No spaces)
PLEASE HELP WILL GIVE BRAINLIEST
Answer:
is it [tex]-2t^{2}[/tex] or [tex]2t^{2}[/tex]?
You are saving for a bike and can save $10 per week. You have $25 when you begin saving. Write an explicit equation
to represent the amount of money you have saved at any given week.
Answer:
your pretty whatsbyour name
Answer:
Amount of money saved 25 35 45 55 65 75 85
Step-by-step explanation:
Explanation: On week 1 you have the $25 you already saved plus the $10 for that week, so $35.
For every week after that add $10 to the money saved for the additional $10 saved that week.
On the 8th week the total amount of money saved is 95+10, or $105.
35x-12= -34x+710 answer fast too
Answer:
x=722/69
Step-by-step explanation:
what are the x and y intercepts of the function f(x)=x/x^2-3x
Answer:x and x2
Step-by-step explanation:
ASAP When 4(0.5x+2.5y-0.7x-1.3y+4) is simplified, what is the resulting expression?
A) -0.8x+4.8y+16
B) 0.8x-4.8y+16
C) -0.8x-4.8y+4
D) 0.8x+4.8y+4
Answer:
The answer is A.
Step-by-step explanation:
4(0.5x+2.5y-0.7x-1.3y+4)
2x+10y+-2.8x+-5.2y+16
2x-2.8x+10y-5.2y+16
= -0.8x+4.8y+16
why cant anyone answer my long math question that has 70 or 60 points on it, come on guys.
Answer:
Math sucks dude.
explanation: math hard
Please answer correctly !!!!!!!!!!!!!!!!!!! Will mark brainliest !!!!!!!!!!!!!!!!!
Combine like terms to write a simplified expression:
12p + 11 - 5p + 3
Answer:
7p + 14
Step-by-step explanation:
add 12p and -5p to get 7p
then add 11 and 3 to get 14
your equation will be 7p + 14
here is the answer
7p + 14.
steve saw 313 birds during a bird-watching trip to Yellowstone National park. he observes 15 more trumpeter swans than sandhill cranes. he saw twice as many mountain chickadees as ducky grouses. how many birds of each type did steve observe if he saw half as many trumpeter swans as ducky grouses
NEED ASAP!
Answer:
Step-by-step explanation:
328