Share
Explore BrainMass

Trigonometry

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 http://commons.wikimedia.org/wiki/File:TrigonometryTriangle.svg.

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.   

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

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

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.

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.

Trigonometric Equation

The number of hours of daylight in city A is given by the following equation, where x is the number of days after January 1. With in a year, when does City A have 12 hours of day light? Give answer in days after January 1. y= 3sin [2Pi/365 (x-79)] +12

Solutions For Five Problems of Mixed Type

Trigonometry/Algebra 1) An open-top box with a square base is to be constructed from 150 square centimeters of material. What dimensions will produce a box with largest possible volume? A) B) C) D) E) 2) Assume h 0. Compute and simplify the difference quotient for f(x) = 1 / x. 3) Solve

Problems with Quadratic Reciprocity & Pythagorean Triangles

2. If p>3, show that p divides the sum of its quadratic residues that are also least residues. (see attached file for diagram) 4. Here is a quadrilateral, not a parallelogram, with integer sides and integer area: (a) What is its area? (b) Such quadrilaterals are not common; can you find another? (c) Could you find 1

Word problems for basic Trig

An airplane travels 100 km/hr for 2 hrs in a direction of 210degrees. At the end of this time how far south of the airport is the airplane? Directions are given in degrees clockwise from north The airplane is -------------kilometers south of the airport after 2 hrs -----------------------------------------------------

Applying the Pythagorean Theorem to Problems

Please help with the following problems involving applications of the Pythagorean Theorem. Provide step by step calculations. 1. The sides of a square are lengthened by 6 cm, the area become 121 cm^2. Find the length of a side of the original square. 2.Television sets. What does it mean to refer to a 20in TV set or a 25in

Vectors, Magnitude, & Direction of a Dart Shot from a Blow Gun

See attached file. Blow gun Using only a meter stick, brains and a calculator, find 1. Vi (coming out of the gun) muzzle velocity, (Hint:first find time) 2. Vfy (a vector) 3. Vf (a vector) When it hit the cardboard, magnitude and direction (Hint: use vfy and

Exemplifying plotting points

Please help with the following problem. Technicians at Buzz Electronics use this equation to determine the amount of current (i) traveling through a circuit at any given time (t): i = 2t^2 â?" 8t + 10 a) Graph the equation to show the relationship between time and current. b) Is this function linear, quadratic, expon

Limits of Trig Functions

** Please see the attached file for the complete solution response ** 2) The limit of f(x) = (sin x)/x as X approaches 0 is 1 a) Let (x, sin x) be a point on the graph of g near (0,0), and write a formula for the slope of the secant line joining (x, sin x) and (0,0). Evaluate this formula at x = 0.1 and x = 0.01. Then

Unit Vectors in Trigonometry

1.) Perform the following operation: a. v + w = ? Where, v = 2i - 6j and w = 3i + 4j b. v - w = ? Where, v = 8i - 6j and w = - 2i + 4j 2.) Find the dot product of vector v and vector w if: a. v = 2i - 3j w = i - j 3.) Find the unit vector that has the same direction as the vector below: a.

Trigonometry Absolute Values for Complex Numbers

1. Find the absolute value of the following complex number: 2. Choose the rectangular coordinates for the following polar coordinate: 3. Determine the rectangular form of the complex number: 4. Find the polar form of the following complex number: 5. When plotted on the rectangular coordinate system in which quadrant

Trigonometry to Determine the Values

1. The graph of a tangent function is given. Select the equation for the following graph: y = tan , y = tan( x +π ), y = tan x, y = tan 2. Graph two periods of the given tangent function. y = 2 tan 2x 3. Graph two periods of the given cosecant or secant function. y = 3 sec x 4.

Calculate the angular speed of the engine in radians per second

1.) The engine of a sports car rotates at 5,000 revolutions per minute (rpm). Calculate the angular speed of the engine in radians per second. Use 2&#61552; radians = 1 revolution. 2.) Convert -60° to radians. Express the answer as a multiple of &#960; 3.) Draw the following angle in standard the position: 4.) In which q

Two trigonometry problems

PLEASE SHOW ALL WORK..... 1. A bicycle tire has a diameter of 20 inches and is revolving at a rate of 10 rpm. At t =0, a certain point is at height 0. What is the height of the point above the ground after 20 seconds? 2. The sun always illuminates half of the moon's surface, except during a lunar eclipse. The illuminated

Trigonometry problems

E= {(x,g(x)): (-3,-7), (-1,-3), (0,-1), (2,3), (4,7)} Write the rule for g-1(x): Show that csc[arccos x] = 1/ root(1-x2): sin 0=.4910; angle 0 = sec(-38 degrees 22') = If F(x) = log5 2x, find F-1(x)._________ Find (a) an upper limit and (b) a lower limit for the zeros of the function P(x)....zeros= -3/2, -r

number of protons, neutrons, and electrons

Parallel Exercises An electron with a mass of 9.11x10-31 kg has a velocity of 4.3 106 in the innermost orbit of a hydrogen atom. What is the de Broglie Wavelength of the electron? An electron wave making a standing wave in hydrogen atom has a wavelength of 8.33 x 10-11 m. If the mass of the electron is 9.11x10-31 kg. what

Trigonometry Resultant Magnitude

1. Two forces P and Q are applied as shown at point A of a hook support. Knowing that P=75 N and Q = 125 N. Determine by trigonometry (a) the required magnitude of the force Q if the resultant R of the two forces applied at A is to be vertical, (b) the corresponding magnitude of R. 2, Solve by trigonometry. Two forces are app

Pythagorean Triangles Programs

Please help me with the following problem: a) 3^2 + 4^2 = 5^2 20^2 + 21^2 = 29^2 119^2 + 120^2 = 169^2 To find another such relation, show that if a^2 + (a+1)^2 = c^2, then (3a+2c+1)^2 + (3a+2c+2)^2 = (4a+3c+2)^2. (b) If a^2 + (a+1)^2 = c^2, let u=c-a-1 and v=(2a+1-c)/2. Show that v is an integer and tha

Pythagoras Theorem: Billboard and Ladder Problem

A billboard painter has been assigned the task of changing the advertisement on a 20 ft billboard, the bottom is 15ft off the ground, two other sites under the sign,one is 10ft from the base the other is 15ft from the base. If the painter uses the patch that is 10ft from the base, what is the length of the ladder required to rea

Distance Formula in 2- and 3-Dimensional Space

Explain how to derive the formula for the distance between two points in analytic geometry in 3-space. (list each necessary step and describe all terms used). Also to discuss how the Pythagorean theorem helps to find distances in 2 and 3 dimensions. Provide at least three examples of how the distance formula can be used in real

History, Definition, and Calculations of Pythagorean Theorem

1) The numbers 3, 4, and 5 are called Pythagorean triples since 3 2+ 4 2= 5 2. The numbers 5, 12, and 13 are also Pythagorean triples since 5 2 + 12 2= 13 2. Can you find any other Pythagorean triples? Actually, there is a set of formulas that will generate an infinite number of Pythagorean triples. 2) Select at least 5 more

Geometry and Trig Questions

Set 1 11: The actual definition of the word tangent comes from the Latin word tangere, meaning "to touch" in mathematics the tangent line touches the graph at a circle at only one point and function values of tan ѳ are obtained from the length of the line segment tangent to a unit circle. Can the line segment ever be greater t