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Mar 28, · Cos 30 Degrees In trigonometry, the cosine function is defined as the ratio of the adjacent side to the hypotenuse. If the angle of a right triangle is equal to 30 degrees, and then the value of cosine at this angle i.e., the value of Cos 30 degree is in a fraction form as v3/2. Jun 04, · Representing Cos 30°: When the ratio of the base of the reference angle with the hypotenuse is taken, it gives us the cosine value of the ratio at that specific angle. When this angle is of 30°, then the value is called cos The value of cos 30 is equal to the value of sin 60 since they complement each other within a triangle.
The Topics Home. The other is the isosceles right triangle. They are special because, with simple geometry, we can know the ratios of their sides. For the definition of measuring angles by "degrees," see Topic Theorem 6. For, 2 is larger than. The cited theorems are from the Appendix, Some theorems of plane geometry. Here are examples of how we take advantage of knowing those ratios.
Example 1. The student should sketch the triangle and place the ratio numbers. Problem 1. To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" "Reload". The sine is the ratio of the opposite side to the hypotenuse.
The tangent is ratio of the opposite side to the adjacent. Problem 2. The cotangent is the ratio of the adjacent side to the opposite. As for the cosine, it is the ratio of the adjacent side to the hypotenuse.
Before we come to the next Example, here is how we relate the sides and angles of a triangle:. If an angle is labeled capital A, then the side opposite will be labeled small a. Similarly for angle B and side bangle C and side c. Example 3. To solve a triangle means to know all three sides and all three angles. Whenever we know the ratios of the sides, we can solve the triangle by the method of similar figures.
And so in how to reduce swelling in ankles when pregnant ABC, the side corresponding to 2 has been multiplied by 5.
Therefore every side will be multiplied by 5. Therefore, side b will be 5 cm. Now, side b is what is the cosine of 30 side that corresponds to 1. And it has been multiplied by 5. It will be 5 how far is staunton va from harrisonburg va. Whenever we know the ratio numbers, the student should use this method of similar figures to solve the triangle, and not the trigonometric Table.
In Topic 6we will solve right triangles the ratios of whose sides we do not know. Problem 3. How long are sides d and f? The student should draw a similar triangle in the same orientation. Then see that the side corresponding to was multiplied by. Therefore, each side will be multiplied by. Side f will be 2. Problem 4. How long are sides p and q? The side corresponding to 2 has been divided by 2. Therefore, each side must be divided by 2.
Problem 5. But this is the side that corresponds to 1. And it has been multiplied by 9. Therefore, side a will be multiplied by 9. It will be 9. Example 4. ABC is an equilateral triangle whose height AD is 4 cm. Solution 1. The height of a triangle is the straight line drawn from the vertex at right angles to the base. Therefore, triangle ADB is a triangle. For this problem, it will be convenient to form the proportion with fractional symbols :.
On taking to be approximately 1. The side corresponding to was multiplied to become 4. How was it multiplied? Problem 6. Example 5. Solve this equation for angle x :. Problem 7. Problem 8. Prove : The area A of an equilateral triangle whose side is s, is.
The area A of any triangle is equal to one-half the sine of any angle times the product of the two sides that make the angle. Topic 2, Problem 6. Problem 9. Prove : The area A of an equilateral triangle inscribed in a circle of radius r, is.
Triangle OBD is therefore a triangle. If we call each side of the equilateral triangle sthen in the right triangle OBD. Problem Prove: The angle bisectors of an equilateral triangle meet at a point that is two thirds of the distance from the vertex of the triangle to the base.
Therefore AP is two thirds of the whole AD. Which is what we wanted to prove. Draw the equilateral triangle ABC. Theorems 3 and 9. From the Pythagorean theorem how to get famous people phone numbers, we can find the third side AD:.
The square drawn on the height of what is the cosine of 30 equalateral triangle is three fourths of the square drawn on the side. Please make a donation to keep TheMathPage online. E-mail: themathpage yandex. Therefore the hypotenuse 2 will also be multiplied by 4 :.
The three radii divide the triangle into three congruent triangles. Theorem 2 Triangle OBD is therefore a triangle. That is what we wanted to prove. We will prove that below. Example 2. You can see that directly in the figure above. BP PD.
Aug 05, · The cosine of 30 degrees is It is also expressed as the square root of three divided by two. The cosine of an angle is calculated by dividing the length of the side of a right triangle adjacent to the acute angle by the length of the hypotenuse. The cosine is a trigonometric function. The exact value of cos(30) cos (30) is v3 2 3 2. v3 2 3 2 The result can be shown in multiple forms. The value of cosine if angle of right triangle equals to 30 degrees is called cos of angle 30 degrees. In Sexagesimal angle measuring system, the cosine of angle 30 degrees is written as cos (30 °).
In mathematics , the trigonometric functions also called circular functions , angle functions or goniometric functions   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.
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.
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.
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.
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. More precisely, the six trigonometric functions are:  . In geometric applications, the argument of a trigonometric function is generally the measure of an angle. However, in calculus and mathematical analysis , the trigonometric functions are generally regarded more abstractly as functions of real or complex numbers , rather than angles.
In fact, the functions sin and cos can be defined for all complex numbers in terms of the exponential function via power series  or as solutions to differential equations given particular initial values  see below , without reference to any geometric notions. The other four trigonometric functions tan, cot, sec, csc can be defined as quotients and reciprocals of sin and cos, except where zero occurs in the denominator.
It can be proved, for real arguments, that these definitions coincide with elementary geometric definitions if the argument is regarded as an angle given in radians. For real number x , the notations sin x , cos x , etc. If units of degrees are intended, the degree sign must be explicitly shown e.
The six trigonometric functions can be defined as coordinate values of points on the Euclidean plane that are related to the unit circle , which is the circle of radius one centered at the origin O of this coordinate system. The trigonometric functions cos and sin are defined, respectively, as the x - and y -coordinate values of point A.
That is,. By applying the Pythagorean identity and geometric proof methods, these definitions can readily be shown to coincide with the definitions of tangent, cotangent, secant and cosecant in terms of sine and cosine, that is. That is, the equalities. The same is true for the four other trigonometric functions. The algebraic expressions for the most important angles are as follows:. Writing the numerators as square roots of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.
Such simple expressions generally do not exist for other angles which are rational multiples of a straight angle. For an angle which, measured in degrees, is a multiple of three, the sine and the cosine may be expressed in terms of square roots , see Trigonometric constants expressed in real radicals.
These values of the sine and the cosine may thus be constructed by ruler and compass. For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the cube root of a non-real complex number. For an angle which, measured in degrees, is a rational number , the sine and the cosine are algebraic numbers , which may be expressed in terms of n th roots.
This results from the fact that the Galois groups of the cyclotomic polynomials are cyclic. For an angle which, measured in degrees, is not a rational number, then either the angle or both the sine and the cosine are transcendental numbers. This is a corollary of Baker's theorem , proved in The following table summarizes the simplest algebraic values of trigonometric functions.
The modern trend in mathematics is to build geometry from calculus rather than the converse. At each end point of these intervals, the tangent function has a vertical asymptote. In calculus, there are two equivalent definitions of trigonometric functions, either using power series or differential equations. These definitions are equivalent, as starting from one of them, it is easy to retrieve the other as a property. However the definition through differential equations is somehow more natural, since, for example, the choice of the coefficients of the power series may appear as quite arbitrary, and the Pythagorean identity is much easier to deduce from the differential equations.
Sine and cosine are the unique differentiable functions such that. Differentiating these equations, one gets that both sine and cosine are solutions of the differential equation. Applying the quotient rule to the definition of the tangent as the quotient of the sine by the cosine, one gets that the tangent function verifies. Applying the differential equations to power series with indeterminate coefficients, one may deduce recurrence relations for the coefficients of the Taylor series of the sine and cosine functions.
These recurrence relations are easy to solve, and give the series expansions . The radius of convergence of these series is infinite. Therefore, the sine and the cosine can be extended to entire functions also called "sine" and "cosine" , which are by definition complex-valued functions that are defined and holomorphic on the whole complex plane.
Being defined as fractions of entire functions, the other trigonometric functions may be extended to meromorphic functions , that is functions that are holomorphic in the whole complex plane, except some isolated points called poles.
Recurrences relations may also be computed for the coefficients of the Taylor series of the other trigonometric functions. These series have a finite radius of convergence. Their coefficients have a combinatorial interpretation: they enumerate alternating permutations of finite sets.
There is a series representation as partial fraction expansion where just translated reciprocal functions are summed up, such that the poles of the cotangent function and the reciprocal functions match: . This identity can be proven with the Herglotz trick.
Similarly, one can find a partial fraction expansion for the secant, cosecant and tangent functions:. For the proof of this expansion, see Sine. From this, it can be deduced that. Euler's formula relates sine and cosine to the exponential function :. This formula is commonly considered for real values of x , but it remains true for all complex values. Solving this linear system in sine and cosine, one can express them in terms of the exponential function:.
When x is real, this may be rewritten as. One can also define the trigonometric functions using various functional equations. For example,  the sine and the cosine form the unique pair of continuous functions that satisfy the difference formula. By taking advantage of domain coloring , it is possible to graph the trigonometric functions as complex-valued functions. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two.
Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities.
For non-geometrical proofs using only tools of calculus , one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function. The cosine and the secant are even functions ; the other trigonometric functions are odd functions. That is:. This means that, for every integer k , one has.
The Pythagorean identity , is the expression of the Pythagorean theorem in terms of trigonometric functions. It is. The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves.
These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula. When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae. These identities can be used to derive the product-to-sum identities. The derivatives of trigonometric functions result from those of sine and cosine by applying quotient rule.
The values given for the antiderivatives in the following table can be verified by differentiating them. The number C is a constant of integration. Alternatively, the derivatives of the 'co-functions' can be obtained using trigonometric identities and the chain rule:.
The trigonometric functions are periodic, and hence not injective , so strictly speaking, they do not have an inverse function. However, on each interval on which a trigonometric function is monotonic , one can define an inverse function, and this defines inverse trigonometric functions as multivalued functions. To define a true inverse function, one must restrict the domain to an interval where the function is monotonic, and is thus bijective from this interval to its image by the function.
The common choice for this interval, called the set of principal values , is given in the following table. As usual, the inverse trigonometric functions are denoted with the prefix "arc" before the name or its abbreviation of the function. When this notation is used, inverse functions could be confused with multiplicative inverses.
The notation with the "arc" prefix avoids such a confusion, though "arcsec" for arcsecant can be confused with " arcsecond ".
Just like the sine and cosine, the inverse trigonometric functions can also be expressed in terms of infinite series. They can also be expressed in terms of complex logarithms. See Inverse trigonometric functions for details. In this sections A , B , C denote the three interior angles of a triangle, and a , b , c denote the lengths of the respective opposite edges. They are related by various formulas, which are named by the trigonometric functions they involve.
The law of sines states that for an arbitrary triangle with sides a , b , and c and angles opposite those sides A , B and C :. It can be proven by dividing the triangle into two right ones and using the above definition of sine. The law of sines is useful for computing the lengths of the unknown sides in a triangle if two angles and one side are known. This is a common situation occurring in triangulation , a technique to determine unknown distances by measuring two angles and an accessible enclosed distance.
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