Trigonometry

General answer: Math

Specific Answer: Taylor Series

Sine, cosine, and tangent all assume you have a right triangle (where one angle is 90 degrees). Each function operates on one number, which represents another angle (call it x) in the triangle. We write these functions as sin(x), cos(x), and tan(x).

Now, we're going to give names to each of the 3 sides. The longest side (the one not touching the right angle) is the hypotenuse. The side touching our angle x, but not the hypotenuse, is the adjacent side. The left-over side is the only one not touching our angle x, which we call the opposite side. We call the length of the hypotenuse h, the length of the adjacent side a, and the length of the opposite side o.

Then sin(x)=o/h

cos(x)=a/h

tan(x)=o/a.

People use the made-up word "sohcahtoa" ("SOAK-a-TOE-ah") to remember this. each third of the word stands for a different function: the letters "soh" stand for "sine" is "opposite" over "hypotenuse", and similarly for "cah" and "toa".

They are co-functions meaning that 90 - sec x = csc x.

To convert from radians to degrees, multiply by 180° and divide by pi.

In all there are [at least] 24 trigonometric functions and ratios. Half of these are circular and the other half are hyperbolic.

Sine and Cosine are basic trigonometric funtions, abbreviated as sin and cos.

Tangent is the third basic ratio defined as Sin/Cos.

For each of these three, there is a corresponding reciprocal function:

Sine -> Cosecant (cosec or csc)

Cosine -> Secant (sec)

Tangent -> Cotangent (cot).

Each of the above six has an inverse function, defined on an appropriate domain. They all are named by adding the prefix "arc", for example arcsin, which is usually written as sin-1.

The above are the circular functions. Each one of them has a corresponding hyperbolic equivalent. These are named by adding the suffix, "h", thus cosh, sech, arccosh [= cosh-1], etc.

sin(t) = 2/3

sin2(t) + cos2(t) = 1

so cos(t) = ± sqrt[1 - sin2(t)]

but because t is in the first quadrant, cos(t) > 0 so cos(t) = + sqrt[1 - sin2(t)]

= sqrt[1 - 4/9] = sqrt[5/9] = sqrt(5)/3

Then sec(t) = 1/cos(t) = 1/sqrt(5)/3 = 3/sqrt(5) = 3*sqrt(5)/5

40 degrees

This is a complicated subject, which can't be explained in a few words. Read the Wikipedia article on "eigenvalue"; or better yet, read a book on linear algebra. Briefly, and quoting from the Wikipedia, "The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix."

Permutations of 10 letters taken 3 a time = 10 x 9 x 8 = 720

1/cos

tan(65 radians) = 1.65, approx.

cos(t) - cos(t)*sin2(t) = cos(t)*[1 - sin2(t)]

But [1 - sin2(t)] = cos2(t)

So, the expression = cos(t)*cos2(t) = cos3(t)

Assuming the wall is vertical, the wall, the ground and the ladder form an isosceles right-angled triangle. Pythagoras tells us that the square of the length of the ladder, in this case 225 equals the sum of the squares of the other two lengths, ie the height where the ladder touches the wall and the bottom of the ladder's distance from the wall. As these distances are equal in an isosceles triangle each must be the square root of (225/2) ie sqrt 112.5 which is 10.6066, as near as makes no difference to 10 ft 71/4 inches

Tan 42 degrees = 0.9004

tan(135) = -tan(180-135) = -tan(45) = -1

In a triangle a/SinA = b/SinB = c/SinC

Since angle A = 43 and angle C is 72, angle B = (180) - (72 + 43) = 65

Hence 20/Sin65 = c/Sin72

20/0.9063 = c/0.9510

c = (20 x 0.9510)/0.9063 = 20.9864

If you are really talking about a closed triangle ABC, then the length of side "a" (given as 19) does not matter in the calculation.

Sum of the angles of a triangle is 180 degrees. Angle B and C add up to 15 + 65 = 80 degrees. Hence angle A is (180 - 80) = 100 degrees

If the 49.5 is in radians, then sin 49.5 ≈ −0.693 and so yes.

If the 49.5 is in degrees, then sin 49.5o ≈ 0.760

If the 49.5 is in gradians, then sin 49.5 ≈ 0.702

If the 49.5 is in some other angle measurement, then you'll have to decide as I only know Degrees, Radians and Gradians angle measures.

In Degrees, one full turn is 360o

In Radians, one full turn is 2π radians ≈ 6.283 radians

In Gradians, one full turn is 400 gradians.

Radians are most useful in calculus.

In fact you've used radians without realising it:

The length of an arc of angle θ of a circle of radius r is θr when θ is measured in radians; the length of an arc of a circle round one full turn (ie the circumference of a circle) is θr = 2πr since one full turn is 2π in radians.

Radius = 5*2/sqrt(3) = 5.77 cm

So area = pi*r2 = 104.72 cm2 (approx)

to simplify Cosx=Sinx Tanx you should remember your fundamental and pythagorean identities..

Cosx + Sinx Tanx

Cosx + Sinx (Sinx/Cosx) <---------- From Tanx= Sinx/Cosx

Cosx + Sin2x/ Cos x <------------- do the LCD

Cosx (Cosx/Cosx) + Sin2x/Cosx

(Cos2x+Sin2x)/Cosx

1/Cosx <--------- From Sin2x + Cos2x =1

or Secx <-------- answer

Comment if you have questions...:))

A function f(x) is even if:

f(x) = f(-x)

In layman's terms this property simply means that any real number in the domain and it's opposite will yield the same function value in the range.

To simplify this down even further, an even function, when graphed will appear to be symetric about the y-axis (assuming that you use the standard Cartesian coordinate plane).

In the case of trig functions, you would have to test whether the even function property holds true for each. We will the test points π, π/2, or π/4. NOTE: The # signs are present next to the functions that are even:

1. Sine: f(x) = sin(x)

-> sin(π/2) = 1, but sin(-π/2) = -1. Since 1 does not equal -1, sine is NOT an even function.

2. #Cosine: f(x) = cos(x)

-> cos(π) = -1 = cos(-π). Since both are equal, cosine IS an even function.

3. Tangent: f(x) = tan(x)

-> tan(π/4) = 1, but tan(-π/4) = -1. Therefore, tangent is NOT an even function.

4. Cosecant: f(x) = csc(x)

-> csc(π/2) = 1, but csc(-π/2) = -1. Therefore, cosecant is NOT an even function.

5. #Secant: f(x) = sec(x)

-> sec(π) = -1 = sec(-π). Since the secant function has asymptotes, it IS an even function provided that x does not equal π(2n+1)/2, where n may be all integers.

6. Cotangent: f(x) = cot(x)

-> cot(π/4) = 1, but cot(-π/2) = -1. Therefore cotangent is NOT even.

f(x) = 1/x except where x = 0.

y=tanx cannot be expressed as a Fourier series, since it has infinite number of infinite discontinuity.

Dirichlet’s condition or the sufficient condition for a function f(x) to be expressed as a Fourier series.

- f(x) is single valued, finite and periodic.

- f(x) has a finite number of finite discontinuities.

• f(x) has a finite number of maxima and minima.

- f(x) has no infinite discontinuity.