Trigonometry

A striaght line , outside the circle, that just touches the circle in one place.

Sin(A) = Opposite/Hypotenuse

Its reciprotcal is

1/Sin(A) = Cosecant(A) = Csc(A) = Hypotenuse / Opposite.

Similarly

Cos(A) = Adjacent/Hypotenuse

Its reciprotcal is

1/Cos(A) = Secant(A) = Sec(A) = Hypotenuse / Adjacent

Tan(A) = Opposite/Adjacent

Its reciprotcal is

1/Tan(A) = Cotangent(A) = Cot(A) = Adjacent / Opposite.

Sine

Its reciprocal is Cosecant

Algebraically Sin ; Reciprocal is '1/ Sin' known as 'Cosecant(Csc)'.

Similarly

Cos(Cosine) ; 1/ Cos (Secant(Sec))

Tan(Tangent) ; 1/ Tan ( Cotangent(Cot)).

Given a complex number z = a + bi, the conjugate z* = a - bi, so z + z*= a + bi + a - bi = 2*a. Note that a and b are both real numbers, and i is the imaginary unit: +sqrt(-1).

There can be no equivalence.

A metre is a measure of length in 1-dimensional space while a metric ton is a measure of mass. The two measure different things and, according to basic principles of dimensional analysis, any attempt at conversion from one to the other is fundamentally flawed.

You find the ( x,y) coordinates of the point where the tangent line just touches the circumference of a circle.

The tangent line will have the straight line eq'n ( y = mx + c)

The circle circumference will have the circle eq'n ( x^(2) + y^(2) = r^(2)

The way to do it is to square up the straight line eq;b/

y^(2) = (mx + c)^(2) = ( (mx)^(2) + 2mxc + c^(2) )

Note how the RHS is squared up.

Subtract to eliminate 'y'

Then you have only 'x' to be solved. Since 'x' is squared 'x^(2)' , and the 'm' , 'r' and 'c' are constants, you can form a quadratic eq'n, to solve for 'x'.

Once solved you substitute 'x' back into either eq'n to find 'y'.

tetrahedron is a 4 faced 3d shape.

Turtle The word turtle is borrowed from the French word tortue or tortre 'turtle, tortoise'. In North America, it may denote the order as a whole. In Britain, the name is used for sea turtles as opposed to freshwater terrapins and land-dwelling tortoises. In Australia, which lacks true tortoises (family Testudinidae), non-marine turtles were traditionally called tortoises, but more recently turtle has been used for the entire group.[4]

The name of the order, Testudines (/tɛˈstjuːdɪniːz/ ⓘ teh-STEW-din-eez), is based on the Latin word testudo 'tortoise';[5] and was coined by German naturalist August Batsch in 1788.[1] The order has also been historically known as Chelonii (Latreille 1800) and Chelonia (Ross and Macartney 1802),[2] which are based on the Ancient Greek word (chelone) 'tortoise'.

Speed = distance / time

S = d/t

For Fritz

S = d/[36/60] hrs

For Traib

S + 48 mph = d / [20/60] hrs

We need to eliminate 'S' , in order to find 'D'.

Hence subtract

S + 48 mph = d / [20/60] hrs

-S = - d/ [36/60] hrs.

48 mph = d/[20/60] - d/[36/60]

Subtraction of fractions.

48 mph = 60d/20 - 60d / 36

48 mph = 3d/1 - 5d/3

48mph = [9d - 5d] / 3

48 mph = 4d / 3

144 mph = 4d

d = 144/4

d = 36 miles/.

Sin(X) = 0.9

X = Sin^(-1) 0.9

X = 64.158... degrees.

Depending on your calculator, you should have an arcsin function, which appears as sin^-1. It's usually a 2nd function of the sin key. If you don't have this function, there are many free calculators you can download... just google scientific calculator downloads.

Anyway, this inverse function will give you theta when you plug in the value of sin theta. Here's the algebra written out:


sin(theta)=-0.0138
arcsin(sin(theta))=arcsin(-0.0138)
theta=.......


The inverse function applied to both sides of the equation "cancels out" the sin function and yields the value of the angle that was originally plugged into the function, in this case theta. You can use this principle to solve for theta for any of the other trig functions:


arccos(cos(theta))=theta
arctan(tan(theta))=theta
and so on, but calculators usually only have these three inverse functions, so if you encounter a problem using sec, csc, or cot, you need to rewrite it as cos, sin, or tan.


sec=1/cos
csc=1/sin
cot=1/tan

Use Cosine Rule

a^(2) = b^(2) + c^(2) - 2bcCosA

Algebrically rearrange

CosA = [a^(2) - b^(2) - c^(2)] / -2bc

Substitute

CosA = [13^(2) - 12^(2) - 5^(2)# / -2(12)(5)

CosA = [ 169 - 144 - 25] / -120

Cos)A) = [0] / -120

CosA = 0

A = 90 degrees (the right angle opposite the hypotenuse)/

However,

If 'A' is the angle between '12' & '13' then 'a' is the side '5'

Hence (Notice the rearrangement of the numerical values).

CosA = [5^(2) - 12^(2) - 13^(2) ] / -2(12)(13)

CosA = [ 25 - 144 -169] / -312

CosA = [ -288[/-312

CosA = 288/312

A = Cos^(-1) [288/312]

A = 22.61986495.... degrees.

To evaluate sin30 plus cos60-2tan45, we start by finding the exact values of sin30, cos60, and tan45.

sin30 = 1/2, cos60 = 1/2, and tan45 = 1.

Substitute these values into the expression: 1/2 + 1/2 - 2(1) = 1/2 + 1/2 - 2 = 1 - 2 = -1.

Therefore, sin30 plus cos60-2tan45 equals -1.

In seismology, trigonometry is used to analyze seismic waves and determine the location and depth of earthquakes. By measuring the time it takes for seismic waves to travel from the earthquake's epicenter to various seismic stations, trigonometric calculations help triangulate the epicenter's position. Additionally, trigonometric functions assist in modeling wave propagation and understanding the angles of wave incidence and reflection, which are crucial for interpreting seismic data.

Inverse of Cosine is 'ArcCos' or Cos^(-1)

The reciprocal of Cosine is !/ Cosine = Secant.

Cos(360 - X) =

Trig. Identity

Cos(360)Cos(x) + Sin(360)Sin(x) =>

1CosX + 0Sinx =>

CosX + o =>

CosX

An octagon is a square with the four corners removed.

These four corners are right angles triangles, with a hypotenuse of of 6 cm. So we need to find the area of these four triangles and remove(subtract) from a larger square.

Using ~Pythagoras.

6^(2) = a^(2) + a^(2)

Hence

36 = 2a^(2)

18 = a^(2)

a = 3sqrt(2) The side length of the triangles.

So the area of these triangles is 4 x 0.5 x 3sqrt(2) x 3sqrt(2( = 36 cm^(2)

Now the side length of a large square enclosing the octagon. is

6 + 3sqrt(2) + 3sqrt(2( = 6 + 6sqrt(2)

The overall area of the 'large square; is ( 6 + 6sqrt(2))^(2) = )Use FOIL).

(6 + 6sqrt(2))(6 + 6sqrt(2)) =

36 + 36sqrt(2) + 36 sqrt(2)+ 72) = 108 + 72sqrt(2) cm^(2)

So subtract from the area , the four corner area which is from above 36cm^(2)

Hence

108 + 72sqrt(2) - 36 = (72 + 72sqrt(2() cm^(2) or

72(1 + sqrt(2)) cm^(2)

or

173.8233765 ... cm^(2).

It is NOT the Law of Sines/Cosine , but the SINE / COSINE Rule.

Sine Rule is SinA/a = SinB/b = SinC/c

Where 'A' , 'B', and 'C' (Capitals) are the angular values.

and 'a', 'b' ,& 'c' (lower case) are the side lengths opposite to the given angle.

For Sine Rule, select any two terms, from the three above. This requires known values for any two angles and one side to find the other side. Or any two sides and one angle to find the other angle.

Known ; angle , angle side to find a side. or Side, side, angle to find an angle. Think 2angles x 2 sides ;

Cosine Rule. is a^(2) = b^(2) + c^(2) - 2bcCosA

This requires three known sides to find an angle. or two known sides and an angle to find the third side.

Think 1 angle x 3 sides.

The Sine ratio is

Sine(angle) = opposite side / hypotenuse side.

This is written in 'short-hand' as

Sin(angle) = o / h

Similarly the cosine ratio is

Cosine(angle) = adjacent side / hypotenuse side.

Cos(angle) = a/h

Similarly the tangent ratio is

Tangent(angle) = opposite side /adjacent side.

Tan(angle) = o/a

NB THe sides refer to the sides of a right-angled triangle.

NNB The angle referred to is NOT the right angle, but a selected angle from the other two angles.

'Theta' is a Classical Greek letter. There is no equivalent in the modern western alphabet.

'Theta' as a letter symbol looks like an oval on its narrow bend, with a bar across its centre.

Remember SecX = 1/CosX

Substitute

SinX X 1 /CosX =

SinX / CosX =

TanX

1

If you mean y = Sin(pi(x))

Then

Use the chain rule

dy/dx = dy/du X du/dx

Let

pi(x) = u

y = Sin (u)

dy/du = Cos(u)

u = pi(x)

du/dx = pi

Combining

dy/dx = pi Cos(u) = piCos (pi(x)). The answer!!!!!

sin(-pi) = sin(-180) = 0 So the answer is 0