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Trigonometric Equations Study Guide

13 Jan 2017Guides

Trigonometric Equations Study Guide

 

Trigonometric Equations

Trigonometry is a branch of mathematics that majorly deals with angles and triangles. Contrary to what most students expect, the subject is not difficult to understand. For starters, passing trigonometry will oblige students to put stock in themselves and encourage themselves that they have what it takes for them to pass. It is conceivable to learn and understand the trigonometric equations in 2 hours if you take the steps below.

For you to understand trigonometric equations, you should sharpen your algebra and geometry skills. In algebra, you should be familiar with changing the subject of equations, solving simultaneous equations and understanding linear equations. Geometry is also close to trigonometry and the areas you need to concentrate on include issues involving circles. Understand the different types of triangles and the different formulas used in solving issues involving triangles.

Keeping in mind the end goal to learn and understand trigonometric equations in 2 hours you should have an understanding of right angled triangles. Compared to other triangles, right angled triangle are the easiest to understand. Purchase learning the basics of these angles; you will have a good understanding of the three ratios in trigonometry. Right angled triangles have three sides, the hypotenuse, adjacent and opposite. The hypotenuse is the longest side of the triangle.

Apart from learning about the angles, you need to understand the three ratios. These are sine, cosine and tangent. The Sine is calculated by dividing the opposite angle by the hypotenuse while the cosine is calculated by dividing the angle adjacent with the hypotenuse. Then again, the tangent is calculated by dividing the angle opposite by the one adjacent. A similar way of arriving at the tangent is to isolate the Sine with the cosine. Below is a summary on each ratio

Sine

The sine function portrays the ratio between the opposite and hypotenuse sides of a triangle. The equation for this relationship is denoted as sin(x) = opposite/hypotenuse. The sine function is an occasional function with a time of 2*pi (or 360 degrees). The sine function oscillates over its domain and has a range of - 1 to 1.

Cosine

The cosine function depicts the relationship between the adjacent and hypotenuse sides of a triangle. The formula is denoted as cos(x) = adjacent/hypotenuse. The cosine function is also an intermittent function with a time of 2*pi (or 360 degrees). The cosine function oscillates just like the sine function, except that it is shifted to the left by 90 degrees. Otherwise, the two graphs seem to be identical in shape.

Tangent

The tangent equation depicts the ratio between the opposite and adjacent sides of a triangle. The equation for this relationship is denoted as tan(x) = opposite/adjacent. Tangent is an intermittent function with a time of pi (or 180 degrees). The tangent function is in fact the division of the two other trigonometric functions and can be denoted as tan(x) = sin(x)/cos(x). Tangent does not resemble the other graphs since it doesn't oscillate in a continuous way. Tan(x) is discontinuous at - pi/2 and pi/2. At this point, the value of tangent is infinite since the value of cos(x) in the quotient is 0. The tangent function has ranged from negative infinity to positive infinite.

Next, learn the basics of other non-right angle triangles. The ratios used in right angle triangles don't play a greater part here but they prove to be useful. For non-right angle triangles, you need to understand two important standards; the cosine and sine run the show. Both of these guidelines are used in determining the properties of non-right-angled triangles. Keep in mind that there are other three ratios that you need to understand. These are the secant, cosecant and cotangent. These are reciprocals of the cosine, sine and tangent respectively.

With regards to trigonometry, you should learn how to comprehend trigonometric equations. These equations usually involve functions and are illuminated by converting the equation and cause it to contain a single ratio. For instance, you can convert a multiple trigonometric functions to contain a single ratio by dividing the equation by a trigonometric term. For example, if the equation contains cosine and sine, isolate it entirely by the Sine. You also need to understand the various trigonometric identities.

Final note

So there are the basics of learning and understanding trigonometric equations. Find some pre-calculus practice issues so you can remember these properties and be ready for differential calculus studies.

Trigonometric Equations Terms and Formulas

Trigonometry, being a branch of mathematics which studies the relationships which exist between the lengths and angles triangles, is replete with a host of terms, fundamental equations and formulas. In this introductory article, we will look at some of the most important trigonometric terms and equations necessary for a student to gain aptitude in trigonometry.

1. Trigonometric Functions

Trigonometric functions are also often referred to as circular functions. They are equations which are functions of an angle. They have a wide range of applications in the study of periodic phenomena including in areas as diverse as navigation, engineering and astrophysics. Indeed, trigonometric functions are fundamental to the understanding of elementary physics especially in the resolution of resolving a vector into Cartesian coordinates. The basic trigonometric functions known as sine and cosine are also central to the understanding of periodic phenomena such as light and sound waves, day length and sunlight intensity as well as temperature variations throughout the year.

Basic Trigonometric Functions

There are six basic trigonometric functions. They are grouped into two:

  • Right-angled triangle derived trigonometric functions
  • Reciprocal functions

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Right-angled Triangle Derived Trigonometric Functions

Trigonometric functions which describe the relationship between the angles and sides of a right-angled triangle are the most basic of all trigonometric functions. A right angled triangle has three sides described according to a chosen angle θ: the adjacent side, the opposite side and the hypotenuse. The three functions which describe this relationship are the Sine, Cosine and Tangent of an angle θ.

Reciprocal functions

The reciprocal trigonometric functions of an angle are derived from the three basic trigonometric functions described above. As implied by the title, reciprocal functions express inverse forms of the three right-angled triangle derived trigonometric functions. They are namely the Cosecant, the Secant and the Cotangent of an angle θ.

2. Domain

A trigonometric domain is a set of all possible inputs of a certain trigonometric function.

3. Range

A trigonometric range is a set of all possible outputs of a stated trigonometric function

4. Inverse Trigonometric Relations

Inverse trigonometric relations are the inverse of the six trigonometric functions sine, cosine, tangent, cosecant, secant, and tangent. The corresponding inverse trigonometric relations of these functions are arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent respectively. If a trigonometric function is used in an equation with two variables, the roles of the two variables will be reversed if the inverse trigonometric function is to have the same meaning. For instance x = sin y has exactly the same meaning as y = arcsin x or y = sin-1x.

5. Inverse Trigonometric Functions

Also often referred to as cyclometric functions, inverse trigonometric functions are inverse trigonometric relations whose range has been highly restricted to ensure there is a one-to-one correspondence between the inputs (numbers) and outputs (angles). As with the basic functions, inverse trigonometric functions have important applications in areas like engineering, astrophysics and navigation. Inverse trigonometric functions are named exactly the same way as the inverse trigonometric relations except that the name of an inverse function begins with an uppercase letter. That means arccosine is an inverse trigonometric relation whose corresponding inverse function is Arccosine.

Formulae Expressing Inverse Trigonometric Relations

  1. x = sin y is same as y= sin-1(x) = arcsine of x = arcsin(x)
  2. x = cos y is same as y= cos-1(x) = arccosine of x = arccos(x)
  3. x = tan y is same as y= tan-1(x) = arctangent of x = arctan(x)
  4. x = cosec y is same as y= cscc-1(x) = arccosecant of x = arccsc(x)
  5. x = sec y is same as y= sec-1(x) = arcsecant of x = arcsec(x)
  6. x = cot y is same as y= cot-1(x) = arctangent of x = arctan(x)

6. Trigonometric Equations

There are two distinct types of trigonometric equations:

  • Trigonometric identities and
  • Conditional equations

Trigonometric Identities

A trigonometric identity is an equation involving trigonometric functions which can be solved by any angle. Indeed, trigonometric identities are much more about evaluating the relationship between functions than they are about evaluation of functions at specific angles.

There are eight fundamental trigonometric identities. From these formative equations, it is possible to formulate an infinite number of derivative trigonometric identities.

If you are to be proficient in trigonometry, you must memorize these eight basic trigonometric identities.

Using knowledge of these eight fundamental identities, you can check and test other identities and even create new ones.

Conditional Equations

Unlike trigonometric identities, a conditional trigonometric equation is an equation which can only be solved by certain specific angles. When solving conditional trigonometric equations, there is one general rule: if there is one solution to the equation, then there are an infinite number of solutions. This counterintuitive state of affairs results from the fact that trigonometric functions are periodic (meaning they repeat for every 36o degrees of arc covered or every 2/7 radians. For instance, if a certain trigonometric has a particular value at 30 degrees, the same will be true at 390, 750, 1110 degrees and so on.

Solving General Equations

Differential equation, for instance dy / dx = x / y, does not comply with a number and function, in this particular case is such that its graph at any point, for example, at coordinates (2,3) has a tangent with slope equal to relative coordinates (in our example 2 / 3).

This is very easily seeing, if we construct a significant number of points and from every delay a short period with a corresponding slope. How to solve differential equations? -The solution would be a function whose graph regards every point of the corresponding segment. If the points and segments enough, we can roughly outline the shape of the curves-solutions (three of these curves are shown in Fig. 1). There's exactly one curve of the answer passing via each point with y - 0. Each individual answer is referred to as a particular solution of differential equations, if we can come across a formula that contains all the partial solutions (except, perhaps, a couple of individuals), then we say that the general solution. Particular answer can be a single function, whereas in popular - their whole family. The best way to solve differential equations - Solve a differential equation - it means to come across either its private or a general solution. In our example, the general solution is y2 x2= c, where c - any number, a particular answer passing via the point (1,1) has the form y = x and is obtained at c = 0; specific solution passing through the point (2,1) has the form y2 x2= 3.

Requiring that the curve is a answer took place, for example, via the point (2, 1), known as the initial condition (as sets the starting point on the curve solving).

y2 x2 =4; y2 x2=0; y2 x2=-4

We can show that inside the example (1) general answer is x = cekt, where c - constant, which may be determined, for example, specifying the quantity of the substance at t = 0.

The equation of Example (2) - a special case of the example (1), corresponding to k = 1 / 100. The initial condition is x = 10 at t = 0 gives the specific solution x = 10et/100. The equation of Example (4) has the general solution T = 70 + cekt along with a specific answer 70 + 130kt; to determine the value of k, more information needed

Differential equation dy/dx = x / y is called the first-order equation; for the reason that it contains the 1st derivative (order differential equation is considered to be the order of entering into it the highest derivative). Most (though not all) emerging practice of differential equations of the first kind pass through each point only one curve is really a solution. You will discover numerous critical varieties of first order differential equations, which admit solutions inside the form of formulas containing only elementary functions - power, exponents, logarithms, sinuses and cosines, etc. These equations are the following.

  • Equation with several variables.
  • Exact differential equations.
  • Linear Equations.
  • The equations of higher orders.
  • Nonlinear differential equations.
  • Existence theorems.
  • Partial differential equations.

Solving linear equations

Examples for Solving Multiple Linear Equations:

Example 1 to solving multiple linear equations:

Find the value of x for the linear equation x+17 =45.

Solution:

The given linear equation is x+17 =45.

Subtract 17 on both sides of the equation to get the value of x.

x +17-17 = 45-17

x = 28

The value of x for the linear equation x+17 = 45 is 28.

Example 2 to solving multiple linear equations:

Find the value of x and y for the equations x+ y = 3 and x- y = 1.

Solution:

The given equations are x+ y = 3 and x- y = 1.

In these equations, first eliminate the value of y.

x+ y = 3 (+) x- y = 1

2x = 4 

Divide by 2 on both sides of the equation.

'(2x)/2' = '4/2'

Solving quadratic equations

Quadratic equation is an equation in which the second power is the highest degree to which the unknown quantity is raised.

The general form of quadratic equation is

ax^2 + bx + c = 0

If ax^2 + bx + c = 0, then the quadratic formula is

x = (-b ± √(b^2- 4ac)) / 2a

Let us learn about the roots of the quadratic equation.

Learn quadratic equations discriminant:

The expression b^2 - 4ac is called discriminant of a quadratic equation. The discriminant of quadratic equations is used to learn the root of quadratic equations.

If b^2 - 4ac > 0, then the roots of quadratic equation are different real numbers.

If b^2 - 4ac = 0, then the roots of quadratic equation are equal real numbers.

If b^2 - 4ac < 0, then the roots of quadratic equation are imaginary numbers.

Example problems to learn how to solve quadratic equations:

A few examples are given below to learn quadratic equations.

Ex 1: Find out the solution for the quadratic equation x2 + 7x + 12 = 0.

Sol : x^2 + 7x + 12 = 0 ............... Given

(x + 4)(x + 3) = 0

x + 4 = 0 or x + 3 = 0

The solution is x = - 4, - 3

Ex 2: How to solve the quadratic equation x^2 - 25 = 0.

Sol : x^2 - 25 = 0 ............... Given

x^2 - 52 = 0

(x + 5)(x - 5) = 0

x + 5 = 0 or x - 5 = 0

x = - 5 or x = 5

The solution is x = -5, 5

Ex 3: What is the solution for the quadratic equation 9x^2 - 6x + 1 = 0.

Sol : 9x62 - 6x + 1 = 0............... Given

9x^2 - 3x - 3x + 1 = 0

3x(3x - 1) - 1(3x - 1) = 0

(3x - 1) (3x - 1) = 0

3x - 1 = 0, 3x - 1 = 0

x = 1/3, x = 1/3

Inverse Trigonometric Relations

In math, cyclometric functions, commonly known as inverse trigonometric functions or arc functions are inverse functions of the various trigonometric functions. These are specifically inverse functions of the tangent, cosine, secant, cotangent, sine and cosecant functions.

 

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Notations

Trigonometric relations can be denoted as in various ways:

  1. The most common one is using the arc prefix. An example is arcsin(x), arctan(x) and arccos(x). Here, C is used to depict the arbitrary constant of integration.
  2. In various computer programming languages, cyclometric functions are denoted as asin, atan, and acos.
  3. The following notations sin-1(x), cos-1(x) and tan-1x were put into use by a scientist John Herschel in 1813. However, this method has been heavily criticized due to its ambiguity.
  4. Another convention commonly used by authors is using the uppercase first letter of the function together with a -1 superscript. For example Sin-1(x) and Cos-1(x).

Basic properties

Since none in the six cyclometric functions is one to one, they are greatly restricted therefore having the inverse functions. Consequently, ranges between inverse trigonometric functions are subsets in the trigonometric functions. An example is, using the function in solving many-valued functions, such as a square root, function y=√x could be defined as y2=x, the following function y=arc sin(x) is explained in a way such that the sin(y) =x.

There are many values(y) whose sine is x. However, when only one resulting value is needed, the function is limited to its principal branch. That is a function which will select only one branch of a multi valued function. With this hindrance, the function will only yield one result, called the principal value. An example is taking the root of a positive real number such as 16 which has two values; +4 and -4. Of these two, positive 4 is picked as the principal value. All the above mentioned properties will usually apply to all six inverse trigonometric functions.

Some writers explain the range of arc secant to be (0≤y≤/2 or π≤y≤3π/2) this is because the tangent part of the function is not negative on the above domain. This ensures consistency and accuracy of computations. For the same reason, writers put the ranges of arc cosecant as -π‹y≤-π/2 or 0‹y≤π/2.

Interrelationships among Trigonometric functions and their inverses.

A quick and fast way of deriving trigonometric functions is by using the geometrical properties of a right angled triangle with a side 1 unit and the other side of any real digit between 0 and 1. After doing this, you apply the Pythagorean relationships and the definitions of trigonometric ratios.

Applications

1. In math

With knowledge of trigonometric inverses, we can calculate the size of angles for example of inclination given certain values in the function. For example.

The base of a ladder is placed 3 meters away from a 10 meter high wall such that the ladder meets top of the wall. What is the measure of the angle made by the ladder and the ground?

Solution: The ladder, wall and ground form a right angled triangle; we therefore use the two known sides in the inverse of tangent function to determine the angle. That is tan-1(10/3)

This is keyed into the calculator while in degree mode and it gives the answer as 73.3 ᶱ

2. In engineering and computer science.

The two argument variant atan2 function calculates the arctangent of y/x given both y and x, however, it must be within a range of –π and π.That means that atan2 is the(lynx) is the angle in between the positive x axis of a straight line plane and any point (x, y) on the plane. These have a positive sign to show counter clockwise angles (y›0) and a negative sign to show clockwise angles (y‹0).These functions are widely utilized in computer programming and engineering set ups.

Numerical accuracy

For angles that are almost close to 0 and π, the arccosine is ill conditioned that is, a small change in the input causes a large variation in the output. Conditionality is a measure of the sensitivity and error in a function. The function will therefore calculate inaccurate values in a computer program implementation because of the limitation in the number of digits. In a similar case, the arcsine is usually inaccurate for angles that are close to –π/2 and π/2. For maximum accuracy in finding all angles, the function of arctangent or atan2 is recommended for use in the implementation.

Inverse Trigonometric Functions

In mathematics, the trigonometric functions are functions of an angle. It is also called as circular function. They can be used in relating the lengths of the sides of a triangle to the angles of a triangle. The most familiar trigonometric functions are the sine, cosine, and tangent. Six trigonometric functions are one-to-one; they must be restricted in order to have inverse function. Integrate the function f(x) with respect to x can be written as'int ' f(x) dx. Every bijective functions has inverse. The function f:R→R defined by f(x) =sinx is not one one, since f(0) = 0=f(π) and hence f is not a bijection. If we restrict the domain and codomain, the function f(x)= sinx may be converted into a bijection. The restricted bijective sine function is denoted by Sinx.

Definition: The function f:[-π/2, π/2] →[-1,1] defined by f(x)=sinx is a bijection. The inverse of f from [-1, 1] into [-π/2, π/2] is also a bijection. This function is called inverse function of Arc sine function. It is denoted by sin-1 or Arc sin.

We find the restricted domain for trigonometric inverse functions

Note 1: If θ belongs to [-'(pi)/(2)' , '(pi)/(2)' ] then Sinθ= sin θ

Note 2: If x belongs to [-1,1] then Sin(Sin-1x)=x

Note 3: If θ belongs to [-'(pi)/(2)' , '(pi)/(2)' ] then Sin-1 (Sinθ) =θ

Note 4: Sin-1x = θ belongs to [-'(pi)/(2)' , 0) iff x = sinθ belongs to [-1,0)

Sin-1x =θ belongs to (0 , '(pi)/(2)' ] iff x = sinθ belongs to(0,1]

Definition: The function f: [0,π]→[-1,1] defined by f(x) = Cosx is a bijection. The inverse of f from [-1,1] into [0, π] is also a bijection. This function is called Inverse cos function or Arc cos function. It is denoted by Cos-1 or Arc cos

Now .Cos-1x= θ iff x = Cosθ, for all x belongs to [-1,1]

Note 1: If θ belongs to[0,'pi' ] then Cosθ= cosθ

Note 2: If x belongs to [-1,1] then Cos (Cos-1x) =x

Note 3: If θ belongs [0,π] then Cos-1 (Cosθ) =θ

Note 4: Cos-1x =θ belongs to [0, '(pi)/(2)' ) iff x= Cos θ belongs to (0,1];

Cos-1x =θ belongs to ( '(pi)/(2)' ,π] iff x= Cos θ belongs to [-1,0);

Definitions of the Inverse Trigonometric Functions - Tan Inverse, Cot Inverse

DEFINITION: -The function f:[-'(pi)/(2)' , '(pi)/(2)' ] →R defined by f(x)=Tan x is a bijection. The inverse of f from R into [-'(pi)/(2)' , '(pi)/(2)' ] is also a bijection. This function is called inverse tan function of Arc tan function. It is denoted by Tan-1 or Arc Tan.

Now Tan-1x =θ iff x =Tan θ for all x belongs to R.

DEFINITION: -The function f: [0,π]→R defined by f(x) = Cot x is a bijection. The inverse of f from R into [0, π] is also a bijection. This function is called Inverse cot function or Arc cot function. It is denoted by Cot-1 or Arc cot.

Now Cot-1x =θ iff x= Cot θ, for all x belongs to R

The domain and range of inverse trigonometric functions are as follows.

Funciton Domain Range

  • Sin-1x [-1,1] [-'(pi)/(2)' , '(pi)/(2)' ]
  • Cos-1x [-1,1] [0,π]
  • Tan-1x R (-'(pi)/(2)' , '(pi)/(2)' )
  • Cot-1x R (0,π) 5 Sec -1x (-∞, -1] U [1,∞) = R - (-1,1) [0, '(pi)/(2)' ) U ('(pi)/(2)' , π]
  • 5 Sec -1x (-∞, -1] U [1,∞) = R - (-1,1) [0, '(pi)/(2)' ) U ('(pi)/(2)' , π]
  • Cosec -1x (-∞, -1] U [1,∞) = R - (-1,1) [-'(pi)/(2)' , 0)U(0, '(pi)/(2)' ]

Solve the Problems on Inverse Trigonometric Functions:

  • Find the value of Sin-1 ([(√5)-1] /4)

Solution:-Given Sin-1 ['(sqrt(5)-1)/(4)' ] = Sin-1 (Sin '(pi)/(10)' ) = '(pi)/(10)'

  • Find the principle values of Sin-1 (Sin '(5pi)/(6)' )= Sin-1 (Sin(π- '(pi)/(6)' ))

= Sin-1 (Sin('(pi)/(6)' ))

= '(pi)/(6)' , Since π/6 belongs to [-'(pi)/(2)' , '(pi)/(2)' ]

Other Trigonometric integration formulas:

  • 'int' '((dx) / (x^2 + a^2)) ' = ' (1/a) tan^(-1)(x / a) ' + c
  • 'int''((dx) / sqrt(a^2 - x^2))' = sin-1'(x / a)' + c

Integration of Inverse Trigonometric Function Problems:

Integration of inverse trigonometric function problem 1:

Find the integration of inverse trigonometric function with respect to x: sin-1 x

Solution:

Given trigonometric function: sin-1x

Take u = sin-1x dv = dx

du = ' 1/sqrt(1-x^2)'dx v = x

'int' sin-1x dx = uv - 'int'v du

= x sin-1x - 'int' x ' 1/sqrt(1-x^2)'dx

Put 1-x2= t therefore, -2x dx = dt

So, 'int'sin-1x dx = x sin-1x -'int' '[((-dt)) /(2sqrtt)]' 

= x sin-1 x + '(1/2)'' [t^(1/2) / (1/2)]' + c

= x sin-1 x + '(1/2) (2/1)' t1/2 + c

= x sin-1 x +' sqrt(1-x^2)' + c

Answer: 'int'sin-1x dx = x sin-1 x +' sqrt(1-x^2)' + c

Integration of inverse trigonometric function problem 2:

Find the integration of inverse trigonometric function with respect to x: cos-1 x

Solution:

Given trigonometric function: cos-1x

Take u = cos-1x dv = dx

du = ' (-1)/sqrt(1-x^2)'dx v = x

'int' cos-1x dx = uv - 'int'v du

= x cos-1x - 'int' x ' (-1)/sqrt(1-x^2)'dx

Put 1-x2= t therefore, -2x dx = dt

So, 'int'cos-1x dx = x cos-1x -'int' '[(-1)((-dt)) /(2sqrtt)]'

= x cos-1 x - '(1/2)'' [t^(1/2) / (1/2)]' + c

= x cos-1 x - '(1/2) (2/1)' t1/2 + c

= x cos-1 x -' sqrt(1-x^2)' + c

Answer: 'int'cos-1x dx = x cos-1 x - ' sqrt(1-x^2)' + c

Integration of inverse trigonometric function practice problems:

Integration of inverse trigonometric function practice problem 1:

Find the integration of inverse trigonometric function with respect to x: tan-1 x

Answer: 'int'tan-1x dx = x tan-1x - '(1/2)' log (x2 + 1) + c

Integration of inverse trigonometric function practice problem 2:

Find the integration of inverse trigonometric function with respect to x: '((1) / (x^2 + a^2)) '

Answer: 'int' '((dx) / (x^2 + a^2)) ' = ' (1/a) tan^(-1)(x / a) ' + cIf you properly follow the above guidelines no Inverse Trigonometric Functions would difficult for you to solve.In case you are stuck you always look online on how to solve Inverse Trigonometric Functions.

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