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Trigonometry is a field of geometry which focuses solely on measuring triangles in terms of the lengths of their sides and their angles. Interestingly, trigonometry first developed for astronomical studies and has evolved to be used for further applications today such as computer graphics and for medical imaging like CAT scans.

All triangles have three angles and three sides. When given a triangle with only one known side value and two known angle values, trigonometry can be used to calculate the missing values. Trigonometry is a field of geometry inexplicitly linked to the measurement of angles.

There are three functions which are integral to the study of trigonometry which are related to the concept of the Pythagorean Theorem. The Pythagorean Theorem is based on the lengths of a right angled triangle’s sides and is represented by the equation:

a2 + b2 = c2. In this equation, c is equal to the hypotenuse, which is the side opposite from the right angle (90 degrees). The values a and b represent one of the other two sides: a is the side opposite of the hypotenuse and b is the side adjacent to the hypotenuse. Figure 1 represents a right angled triangle. 

Figure 1. This figure is representative of a right angled triangle and the variables labelled on this triangle correspond with the Pythagorean Theorem and the three trigonometric functions. The lower case letters represent side length values and the capitalized letters represent angle values. This image has been taken from

The three functions integral to the study of trigonometry are as follows (the ratios use the lengths of the sides):

  1. Cosine Function: This function has the following ratio:

cos A = side adjacent to the hypotenuse/hypotenuse (a/h)

  1. Sine Function: This function takes the following ratio:

sin A = side opposite from the hypotenuse/hypotenuse (o/h)

  1. Tangent Function: This function utilizes the following ratio:

tan A = side opposite the hypotenuse/side adjacent to the hypotenuse (o/a)

The mnemonic SOH-CAH-TOA is often taught as a way to remember which function matches with which ratio.   

Solving Triangular Equation in Satellite Example

A satellite circles the earth in such a manner that it is y miles from the equator (north or south, height from the surface not considered) t minutes after its launch, where y=5000[cos( π/45)(t-10)]. At what times t in the interval [0,240], the first 4 hours is the satellite 3000 mi north of the equator?

A Series of Trigonometry Problems

I have some trig questions I need help with: 1. find the number of degrees in the measure of the smallest positive angle that satisfies the equation 2cos(x) + 1 = 0 2. In the interval 0 degrees < (or equal to) x < (or equal to) 360 degrees, sin (x) = -1. Find the number of degrees in the measure of angle x 3. If x is

Parametric Representation of a Curve

See the attached file. 1. Find a parametric representation of the curve: x^2 + y^2 = 36 and z = 1/π arctan (x/y) i.e find a representation in the form: x = x(t); y=y(t), z = z(t) 2. Find what kind of curves are given by the following representations and draw (schematically) the curves: i) r(t) = (2t - 5, -3t + 1,4):

Using the Laws Related to Solving Oblique Triangles

Solving oblique triangles: The trigonometry of oblique triangles is not as simple as that of right triangles, but there are two theorems of geometry that give useful laws of trigonometry. These are called the "law of cosines" and the "law of sines." There are other "laws" that used to be used, but since the common use of calcul

Trigonometry Proofs

Question: Why does abs (pi-4) = 4 - pi ? ( I think it's because pi -4 = -4+pi = -(4 - pi) and then abs (-(4 - pi)) = 4-pi, but I am not too sure). But here is what I really want to know, my book says abs (7 - pi) = 7- pi. If the above is true shouldn't it = pi - 7?

Trigonometry: Polar and Rectangular Coordinates

Convert the equation to polar coordinates and simplify. 1. a. xy = 1 b. (x^2 + y62)^2 = 2xy Express the equation in rectangular coordinates. 2. a. r = 3sin(t) b. r = 4cos3(t) r^2 = 4sin2(t) r = 4/(2 + cos(t))

Trigonometry Simplification

Use an addition of subtraction formula to simplify the equation. Then find all solutions in the interval [0, 2*pie). a) cos(x)*cos(lambda)(x) + sin(x)*sin(2x) = 1/2 b) sin(3x)*cos(x) - cos(3x)*sin(x) = 0.

Applying the Pythagorean Theorem

MY QUESTION: DO I USE THE PYTHAGOREAN THEOREM TO SOLVE FOR THE DISTANCE? (Please note that the all Caps used, to distinguish my question from the question for the assignment below). Meagan has just moved into a house very close to her mom and her sister. She is ready to move her belongings that are stored at her mom's house

Trigonometry Finding Exact Values

1. Find the exact value of the following expressions given that sec(x) = 3/2, csc(y) = 3 and x and y are in quadrant I. a) cost(x - y) b) sin(2y) c) tan(y/2) 2. Write the following expressions in terms of sin only: a) sin(x) + cos(x) b) 3sin(pi*x) + 3*sqrt(3)*cos(pi*x)

Trigonometry Problems and Identities

Please see the attachment for the exact notations and questions in their original format. Basically, what I need help with is the following: 1) Use formulas for lower powers to rewrite the expression in terms of the first power of cosine. a) cos (^4)x b) (cos (^4)x) (sin (^2)x) c) cos (^6)x 2) Prove the identity a)

Trigonometric Methods

Please assist in supplying fully worked out answers to the problems showing all calculations, transformations of formulas and a brief summary of the work so that I can use these towards further examples. See the attached file for the problems.

Using Trigonometric Identities

1. Use the fundamental identities to write the first expression in terms of the second: a) tan^2(t)*sec(t); cost(t) b) sec(t); sin(t) with t in quadrant 2 Find the values of the remaining trigonometric functions at t from the given information: a) If sin(t) = -8/17 and the terminal point for t is in quadrant 4, find csc(t)

Describe the Variation in Water Level as a Function

Please help me with this trigonometry question: The graph shows the variation of the water level relative to mean sea level in Commencement Bay at Tacoma, Washington, for a particular 24-hour period. Assuming that this variation is modelled by simple harmonic motion, find an equation that describes the variation in water leve

Find the Graph of a Sine or Cosine Curve

The graph of one complete period of a sine or cosine curve is given in the attached document. a) Find the amplitude, period and phase shift. b) Write an equation that represents the curve in the form: y = a sin k (x - b) or y = a cos k (x - b)

d'Alembert's solution

Consider the wave equation for a semi-infinite string(in the domain x>or =0) with wave speed c=1, for initial conditions u(x,0)=0 and (u) subscript (t)(x,0)= (4x)/(1+x^2), x>or =0 Using d'alembert's solution show that the solution of the wave equation for t>or=0 is u(x,t)=In((1+(x+t)^2)/(1+(x-t)^2)) I have to consider t

Sinus cosinus functions

A merry-go-round takes 15 seconds to complete one revolution/spin. Within that time, each horse moves up and down five times. The vertical motion of the horse spans a range of 50 cm, and the horse is 1 m high at its vertically centre position. (Hint: Sine Wave/Function) a. Sketch a graph of the height of the horse over time fo

Trigonometry: Application in a Ferris Wheel

Please help with the following problem involving a trigonometry application. A ferris wheel has a radius of 26 ft and makes one revolution counterclockwise every 12 sec. If t=0 represents the 6 o' clock position, find a formula to represent the height of a person on the ferris wheel after t seconds.

Verify the trigonometry identities

The trigonometry identities to be proved: 1) cos(4x) = 1 - 8sin^2(x) + 8sin^4(x) 2) tan(4x) = [4tan(x) - 4tan^3(x)]/[1 - 6tan^2(x) + tan^4(x)] 3) 2cos(4x)sin(2x) = 2sin(3x)cos(3x) - 2sin(x)cos(x) 4) 2sin(x)sin(2x) = 4cos(x)sin^2(x) 5) cos(x+y)cos(x-y) = cos^2(x) + cos^2(y) - 1 6) cos(x+y)sin(x-y) = sin(x)cos(x) - sin(y)c

Verifying the Identity of Trigonometry

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Computation of the First Two Pythagorean Triples

Generate two Pythagorean triples. After building them, verify each of them in the Pythagorean equation. Please include all math, including steps and formulas used to solve the problem, and why you selected that method to solve the problem along with a summary of the problems.

Using Trigonometry to Solve the Question

A ship captain at sea uses a sextant to sight an angle of elevation of 37 degrees to the top of lighthouse. After the ship travels 250 feet directly toward the lighthouse, another sighting is made, and the new angle of elevation is 50 degrees. The ship charts show that there are dangerous rocks 100 feet from the base of the ligh

Derivatives of the sine and cosine functions

The voltage (V) in a certain electric circuit as a function of the time (t) in seconds is given by 3.0 sin184t cos184t. How fast is the voltage changing when t=0.002 seconds? The answer will look like 412 V/s when t=2.0 ms ?

Graph Analysis for Trigonometry

In a baby's room, a toy is suspended from a bar that hangs over the baby's crib. When the baby pulls and releases the toy, it begins to bob up and down so that the distance between the toy and the bar oscillates in a sinusodial fashion. Let d(t) be the toy's distance from the overchanging bar, measured in centimeters, after t se

Law of sines

Two stations are 190 miles apart, station A directly south of station B. Both spot a fire. The bearing of the fire from station A is N55degreesE and the bearing of station B is S60degreesE. How far, to the nearest tenth of a mile, is the fire from each lookout station. Distance from staion B to the fire is: Distance from s

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Trigonometry Word Problem to Solve for Distance

In the land of Lilliputian, Michelle and Sean was boat riding of the coast of Coackatoo. Well, they almost ran into the buoy and ended up capsizing their boat. The rescue team came to rescue them but they could only fly over the buoy. But Michelle and Sean were floating quite a distance away from the buoy with Michelle closest t

Trigonometric Equation

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converse of the Pythagorean theorem

Please give an example of a mathematical statement whose converse is not true. Prop. 48: If in a triangle the square on one of the sides be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right.

Pythagoras theorem for airplane flights

An airplane flies over an observer with a velocity of 400 mph and at an altitude of 2640 ft. If the plane flies horizontally in a straight line, find the rate at which the distance x from the observer to the plane is changing 0.6 min after the plane passes over the observer.