Trigonometric Functions Formula Definitions and Examples
Trigonometry is the branch of mathematics that deals with the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words for “triangle” and “measure.” There are six trigonometric functions, which are usually denoted by the letters sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). These six trigonometric functions inverses, angles, and identities are essential to understand when studying trigonometry. In this blog post, we will explore these topics in depth with definitions, formulas, and examples.
Trigonometry Formulas
There are numerous trigonometry formulas that are used to solve various problems in mathematics. The most commonly used trigonometry formulas are:
-Pythagorean Theorem: a^2 + b^2 = c^2
-Distance Formula: d = (x_2 – x_1)^2 + (y_2 – y_1)^2
-Law of Sines: a/sinA = b/sinB = c/sinC
-Law of Cosines: a^2 = b^2 + c^2 – 2bc cosA
List of Trigonometry Formulas
Trigonometric functions are defined as the ratios of the sides of a right angled triangle. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan). These functions have many applications in mathematics, physics, engineering, and other sciences.
There are a number of different trigonometry formulas that can be used to solve various problems. The following is a list of some of the more commonly used formulas:
Sine function: sin(x) = opposite side/hypotenuse
Cosine function: cos(x) = adjacent side/hypotenuse
Tangent function: tan(x) = opposite side/adjacent side
Pythagorean theorem: a^2 + b^2 = c^2
Angle addition formulas: sin(x+y) = sin(x)cos(y)+cos(x)sin(y) and cos(x+y)= cos(x)cos(y)-sin(x)sin(y)
Basic Trigonometry Formulas
Basic Trigonometry Formulas
Sin, cos and tan are the basic functions of trigonometry. The sine function, denoted by sin(?), is defined as the ratio of the length of the side opposite to the angle ? to the length of the hypotenuse. The cosine function, denoted by cos(?), is defined as the ratio of the length of the side adjacent to the angle ? to the length of the hypotenuse. The tangent function, denoted by tan(?), is defined as the ratio of the length of the side opposite to angle ? to that of its adjacent side.
The following are some basic trigonometry formulas:
1. sin2? + cos2? = 1 ……………………….(identity)
2. cosec2? = 1 + tan2? ……………………. (reciprocal identity)
3. sec2? = 1 + cot2? ……………………… (reciprocal identity)
4. cosec ? = 1/sin ? ……………………. (definition)
5. sec ? = 1/cos ? …………………….. (definition)
6. tan ? = 1/cot ? ……………………. (definition)
Trigonometry Formulas Involving Reciprocal Identities
There are a few trigonometry formulas that involve reciprocal identities. These are identities that you can use to simplify expressions that contain reciprocals. Here are some examples:
1. The reciprocal of sine is cosecant:
The reciprocal of cosine is secant:
2. The reciprocal of tangent is cotangent:
3. The reciprocal of cosecant is sine:
4. The reciprocal of secant is cosine:
5. The reciprocal of cotangent is tangent:
These relationships between the reciprocals of trigonometric functions can be useful when you’re simplifying expressions or solving equations. For example, if you need to find the value of sin15°, you could use the fact that sin15° = 1/2 and then use the reciprocal identity for sine to write it as 2sinx=1. Then you can solve for x using basic algebra and find that x=30°.
Trigonometric Ratio Table
Trigonometric ratios are the ratios of the sides of a right angled triangle. The ratios are independent of the size of the triangle.
The trigonometric ratios for a right angled triangle are:
Sine (sin) = opposite/hypotenuse
Cosine (cos) = adjacent/hypotenuse
Tangent (tan) = opposite/adjacent
Trigonometry Formulas Involving Periodic Identities(in Radians)
1. sin(x+y) = sin x cos y + cos x sin y
2. cos(x+y) = cos x cos y – sin x sin y
3. tan(x+y) = tan x + tan y / 1 – tan x tan y
4. cot(x+y) = cot x – cot y / 1 + cot x cot y
5. sec(x+y) = sec x + tan x sec y / 1 + tan x tan y
6. csc(x+y) = csc x – cot x csc y / 1 + cot^2 xtan^2 y
Trigonometry Formulas Involving Co-function Identities(in Degrees)
There are a few trigonometry formulas involving co-function identities that are important to know when working with degrees. These include:
– sin(90° – x) = cos(x)
– cos(90° – x) = sin(x)
– tan(90° – x) = cot(x)
Each of these formulas can be derived from the basic definition of a co-function: f(x) = 1/f(?/2 – x). Using this definition, it is easy to see how the above formulas work.
For example, let’s look at the first formula: sin(90° – x) = cos(x). We can plug in our known values and rearrange to get: cos(x) = 1/sin(?/2 – x). From here, we can substitute ?/2 for 90° and simplify to get the final result.
Trigonometry Formulas Involving Sum and Difference Identities
When working with trigonometric functions, there are a few basic identities that you should be familiar with. These identities involving sums and differences can be very useful when simplifying expressions or solving equations.
Here are the most basic sum and difference identities:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
cos(x + y) = cos(x)cos(y) – sin(x)sin(y)
tan(x + y) = tan(x) + tan(y)/[1 – tan(x)*tan(y)]
These identities can be derived from the more fundamental addition formulas for sine and cosine:
sin(x + y)= sin x cos y + cos x sin y
cos (x+y)= cos x cos y – sin x sin y
and can be used to derive many other trigonometric identities.
Trigonometry Formulas For Multiple and Sub-Multiple Angles
The Trigonometry Formulas For Multiple and Sub-Multiple Angles are:
For any angle A,
sin(2A) = 2sinAcosA
cos(2A) = cos^2A – sin^2A
tan(2A) = 2tanA/(1 – tan^2A)
These formulas can be used to find the values of trigonometric functions for angles that are multiples or sub-multiples of other angles. For example, if you know the value of sin60°, you can use the formula for sin(2A) to find the value of sin120°.
Trigonometry Formulas Involving Half-Angle Identities
Trigonometry is the branch of mathematics that studies relationships between angles and sides of triangles. The most basic trigonometric functions are sine, cosine, and tangent. These functions have many applications in science and engineering.
The half-angle identities are a set of trigonometric identities that involve taking the sine, cosine, or tangent of half an angle. These identities can be used to simplify expressions and to solve problems.
Here are some examples of half-angle identities:
sin(?/2) = ?( (1 – cos(?)) / 2 )
cos(?/2) = ?( (1 + cos(?)) / 2 )
tan(?/2) = 1 / ?( (1 – cos(?)) / (1 + cos(?)) )
These identities can be used to simplify expressions involving trigonometric functions. For example, if you want to find the sine of an angle ?, you can use the identity sin(?) = 2sin(?/2)cos(?/2). This can be helpful if you know the value of sin(?/2) but not sin(?).
Trigonometry Formulas Involving Double Angle Identities
To start with, let’s recall the definition of a trigonometric function. A trigonometric function is a function that takes an angle as input and returns a ratio of sides of a right triangle. The most common trigonometric functions are sine, cosine, and tangent, although there are many more.
Now let’s move on to double angle identities. A double angle identity is an identity that is true for all angles. For example, the following identity is true for all angles x:
sin(2x) = 2sin(x)cos(x)
This identity can be proven by using the Pythagorean Theorem. We will not go into the proof here, but it is something you can explore on your own if you’re interested.
There are many other double angle identities involving other trigonometric functions, such as cosine and tangent. Some of these identities are listed below:
cos(2x) = cos^2(x) – sin^2(x) = 2cos^2(x) – 1
tan(2x) = 2tan(x)/(1 – tan^2(x))
sin^2(x) + cos^2(x) = 1
Trigonometry Formulas Involving Triple Angle Identities
The most important thing to remember when working with trigonometric functions is that they are periodic. This means that they repeat themselves over and over again. The period of a function is the amount of time it takes for the function to repeat itself. For example, the cosine function has a period of 2pi radians, which means it will repeat itself every 2pi radians (or 360 degrees).
There are a number of different identities involving triple angles that you should be aware of when working with trigonometric functions. These identities can be used to simplify equations or to help you solve problems.
Some of the most important triple angle identities are:
cos(3?) = 4cos^3(?) – 3cos(?)
sin(3?) = 3sin(?) – 4sin^3(?)
tan(3?) = 3tan(?) – 4tan^3(?)
cot(3?) = 3cot( ?)- 4cot^3(?)
Trigonometry Formulas – Sum and Product Identities
In mathematics, trigonometric identities are equations that involve trigonometric functions and are true for all values of the occurring variables where both sides of the equality are defined. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.
There are an infinite number of trigonometric identities, but there are a few that are particularly useful, including the sum and product identities. In this section, we’ll take a look at these identities and some examples of how to use them.
The sum identity states that for any angles A and B, the following equation holds true:
sin(A + B) = sinAcosB + cosAsinB
This identity can be used to simplify expressions that involve the sine of a sum of angles. For example, consider the expression sin(30° + 45°). Using the sum identity, we can rewrite this as follows:
sin(30° + 45°) = sin30°cos45° + cos30°sin45°
= 0.5cos45° + 0.866sin45°
= 0.7378640776699029…
Similarly, the product identity states that for any angles A and B, the following equation holds true:
sinAcosB = 1/2[cos(A – B) – cos(A + B)]
This identity can be used to
Trigonometry Formulas Involving Product Identities
There are several trigonometry formulas involving product identities that are useful for solving problems. These include the sine and cosine of a sum or difference, the double-angle formulas, and the half-angle formula.
The sine and cosine of a sum or difference can be found using the following formulas:
sin(A+B) = sinAcosB + cosAsinB
sin(A-B) = sinAcosB – cosAsinB
cos(A+B) = cosAcosB – sinAsinB
cos(A-B) = cosAcosB + sinAsinB
The double-angle formulas can be used to find the sine or cosine of twice an angle. These formulas are derived from the half-angle formulas below.
sin(2A) = 2sinAcosA
cos(2A) = cos^2 A – sin^2 A OR 1 – 2sin^2 A
Finally, the half-angle formula can be used to find the sine or cosine of half an angle. These formulas can be derived from the double-angle formulas above.
sin(A/2) = +/- sqrt((1+cosA)/2) OR +/- (sqrt(1-sin^2 A)/cos A) // sign is positive if A in Quadrant I or IV,
Trigonometry Formulas Involving Sum to Product Identities
There are a variety of trigonometry formulas involving sum to product identities that can be used to solve various problems. These formulas can be used to simplify equations, solve for unknown angles or sides of a triangle, and more. Some of the most commonly used trigonometry formulas involving sum to product identities are listed below:
-Sum to Product Identity for Sin: sin(A + B) = sinAcosB + cosAsinB
-Sum to Product Identity for Cos: cos(A + B) = cosAcosB – sinAsinB
-Difference Formula for Sin: sin(A – B) = sinAcosB – cosAsinB
-Difference Formula for Cos: cos(A – B) = cosAcosB + sinAsinB
-Double Angle Formula for Sin: sin2A = 2sinAcosA
-Double Angle Formula for Cos: cos2A = cos2A – sin2A
Inverse Trigonometry Formulas
Inverse trigonometric functions are used to solve problems involving angle measurement in radians or degrees. These functions allow us to calculate the size of an angle without knowing the length of the sides of the triangle. The most common inverse trigonometric functions are:
– Arccosine (acos)
– Arcsine (asin)
– Arctangent (atan)
– Cotangent (cot)
– Secant (sec)
– Sine (sin)
– Tangent (tan)
Trigonometry Formulas Involving Sine and Cosine Laws
The sine and cosine laws are two trigonometric formulas that allow us to solve for unknown sides and angles in a triangle. These laws are essential for anyone studying trigonometry, as they provide a way to calculate all sorts of properties about triangles.
The sine law states that:
sin(A)/a = sin(B)/b = sin(C)/c
where A, B, and C are the angles of the triangle, and a, b, and c are the lengths of the corresponding sides. So, if we know two angles and one side length in a triangle, we can use the sine law to solve for the other two side lengths. For example, if we know that angle A is 60 degrees, angle B is 30 degrees, and side a is 10 units long, we can solve for side b and side c using the following equations:
sin(60)/10 = sin(30)/b ~~> b = 5 units
sin(60)/10 = sin(C)/c ~~> c = 10/sin(60) ~~> c = 16.97 units
What are Trigonometric Functions?
A trigonometric function is a function that relates an angle to a ratio of two sides of a right triangle. The most common trigonometric functions are sine, cosine, and tangent.
The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.
These ratios can be used to solve problems involving triangles, such as finding unknown lengths or angles. Trigonometric functions can also be graphed, and these graphs can be used to solve problems involving periodic motion.
The Different Types of Trigonometric Functions
There are three common trigonometric functions: sine, cosine, and tangent. Each of these functions has a specific definition and formula, which is used to calculate different properties of angles.
Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
The Formula for Trigonometric Functions
When we talk about trigonometric functions, we’re referring to a group of functions that helps us understand relationships between angles and sides in triangles. The most common trigonometric functions are sine, cosine, and tangent, although there are others (such as cosecant, secant, and cotangent). These functions can be represented using the following formulas:
Sine (sin): sin(?) = opposite / hypotenuse
Cosine (cos): cos(?) = adjacent / hypotenuse
Tangent (tan): tan(?) = opposite / adjacent
These formulas allow us to calculate the values of these functions for any given angle. For example, if we know that the angle ? is equal to 30°, we can plug that value into each of the above formulas to find the corresponding function values:
sin(30°) = opposite / hypotenuse
=> sin(30°) = 0.5 / 1
=> sin(30°) = 0.5
cos(30°) = adjacent / hypotenuse
=> cos(30°) = ?3/2 / 1
=> cos(30°) = ?3/2
tan(30°) = opposite / adjacent
=> tan(30°) = 0.5/?3 / 1
=> tan(
Conclusion
Trigonometric functions are very important in mathematics and have many applications in physics and engineering. In this article, we have provided a brief introduction to trigonometric functions, their formulas, and some examples. We hope that you have found this information to be helpful and that you will use it to further your understanding of these important concepts.
Examples of Trigonometric Functions
There are many trigonometric functions, each defined as the ratio of two sides of a right triangle. The most common trigonometric functions are sine, cosine, and tangent, but there are also secant, cosecant, andcotangent. These six functions are abbreviated as sin, cos, tan, sec, csc, and cot.
The sine function is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. The cosine function is defined as the ratio of the length of the side adjacent to the angle to the length ofthe hypotenuse. The tangent function is defined as the ratio of the length of the side opposite to the angle to the length ofthe side adjacent to the angle.
To find secant, cosecant, or cotangent, we take the reciprocal (inverse) of sine, cosine, or tangent respectively. So secant is 1/sine, cosecant is 1/cosine, andcotangent is 1/tangent.
Here are some examples:
Sin(30°)=0.5 because in a 30-60-90 right triangle (a special type of right triangle where all angles measure 60° or 90°),the length ofthe side opposite 30° is half thatofhypotenuse Cos(45°)=
