Trigonometry

To determine the length of BC, we need more information about the relationship between points A, B, C, D, E, and F, such as the configuration of these points (e.g., are they on a straight line, a triangle, etc.). Without additional context or a diagram, we cannot calculate the length of BC based solely on the provided lengths of AB, AC, DE, DF, and EF.

-Sin^(2)(Theta) + Cos^(2)Theta =>

Cos^(2)Theta - Sin^(2)Theta

Factor

(Cos(Theta) - Sin(Theta))( Cos(Theta) + Sin(Theta))

#Is the Pythagorean factors .

Or

-Sin^(2)Theta = -(1 - Cos^(2)Theta) = Cos(2)Theta - 1

Substitute

Cos^(2)Thetqa - 1 + Cos^(2) Theta =

2Cos^(2)Theta - 1

Tan(60) = Sin(60)/ Cos(60)

Sin(60) = sqrt(3)/2

Cos(60) = 1/2

Hence

Sin(60) / Cos(60) = [sqrt(3) / 2] / [1/2} =>

sqrt(3) / 2 X 2/1 sqrt(3)

Hence Tan(60) = sqrt(3) = Numerically = 1.732050808....

Sine and cosine are both trigonometric functions that relate to angles in a right triangle, but they represent different ratios. The sine function, denoted as sin(θ), gives the ratio of the length of the opposite side to the hypotenuse, while the cosine function, denoted as cos(θ), gives the ratio of the length of the adjacent side to the hypotenuse. In the unit circle, sine corresponds to the y-coordinate and cosine corresponds to the x-coordinate of a point on the circle at a given angle. This fundamental difference leads to distinct properties and applications in various fields such as physics, engineering, and mathematics.

NO!!!

The sum of the two shorter sides MUST exceed the longest side.

1 + 2 = 3 ; 3 So the sum does not satisfy the criteria.

a triangle-based pyramid

Is the '1' one degree , or one radian or, Tan(angle) = 1.

You need to clarify.

However,

Tan(1 degree) = 0.017455...

Tan( 1 radian) = 1.55740....

Tan(angle) = 1

Angle = Tan^(-1) 1 = 45 degrees. = pi/4 radians.

Tan(35) = 0.700207....

However,

Tan = Sin/ Cos

Hence

Tan(35) = Sin(35) / Cos(35) = 0.57357... / 0.81915...

&

0.57357... / 0.81915.... = 0.7002.... As before!!!!!

The area of a right angle is nothing bro, if you mean the area of a right angle triangle then uts simple formula is : 1/2×Base×Height(Perpendicular of the right angled triangle)

Alternative method:

Heron's formula!

A Tangent is a straight line, outside the circle, that just touches the circle circumference in one place.

A Chord is a straight line , inside the circle, but not passing throught the centre. and touches the circle circumference in two place.

First make sure your calculator is in 'Degree Mode (D)'.

Then using the 'Inverse' of 'Sin' , shown as 'ArcSin' or ' Sin^(-1)' . enter '0.5', followed by '=' . The answer should be '30' ( 30 degrees).

In 2-dimension it is an 'HEXAGON'

In 3 dimension is is am HEXAHEDRON.

The correct name for a 3D triangular pyramid is a 'TETRAHEDRON. It has FOUR(4) faces.

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.