Trigonometry

P(First card) = 13/52

P(second card = 12/51 Because you have already drawn one spade , there is one less card in the pack .

P(Both spades) = 13/52 X 12/51 = 156/2652 = 0.0588

Cos(2A) = Cos(A + A)

Double Angle Indentity

Cos(A+A) = Cos(A)Cos(A) - Sin(A)Sin(A) =>

Cos^(2)[A] - SIn^(2)[A] =>

Cos^(2)[A] - (1 - Cos^(2)[A] =>

2Cos^(2)[A] - 1

On a calculator type in 'Sin' , followed by '30' follwed by '=' . The answer is '0,5'.

Sin(30) = 0.5

All the trig. values are based on a triangle integrated into a circle.

Practically,

On a piece of graph paper, draw a circle centred on the origin (0,0).

Call the radius of that circle '1'.

Now draw a radius 30 degrees from the x-axis.

Then drop a line from point on the circle were the radius meets( intersects) the circle. to the x-axis, to meet the x-axis at 90 degrees. This 'dropped' line is the 'opposite' to the angle 30 degrees. Measure the length of this 'dropped'; line. You will find it to be exactly half (0.5 = 1/2) the length of the radius.

Similarly, if the radial angle is 45 degrees a corresponding sine value is 0.707106781... = sqrt(2)/2 = 1/sqrt(2).

All other 'Sine' values can be found in a similar manner. However, to save time on drawing circles and triangles, the trig. values are integrated into modern calculators or formerly into Castle's Four Figure Tables.

This is NOT a Pythagorean triangle.

So we must fall back on the Cosine Rule for an agular value.

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

Algebraically rearranging

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

Substituting

CosA = [10^(2) - 7^(2) - 5^)2)] / -(2(7)(5))

CosA = [100 - 49 - 25]/-70

CosA = 26/-70

CosA = -0.371428571...

A = Cos^(-1) -0.371428573

A =111.8037 481 ... degrees. -

Now area of triangle is 0.5 X Base X height

The height is = 7Sin(111.8037481...)

Hence are is 0.5(5)(7)Sin(111.8037481...)

Area = 16.248... m^(2)

The Arc of a circle is that part of the circumference between two radii.

Sin(theta) = 0.03125

Hence

theta = ArcSin(0.03125)

theta = 1.790784659... degrees.

There is no other names!!!!

Howver 'the word' trigonmetry' comes from Latin, and means 'measuring triangles.'.

trigon ( a three sided 2-dimenstional figure) = triangle (modern).

Metry = to measure.

Tan(x) = Sin(x) / Cos(x)

Hence

Sin(x) / Cos(x) = Cos(x)

Sin(x) = Cos^(2)[x]

Sin(x) = 1 - Sin^(2)[x]

Sin^(2)[x] + Sin(x) - 1 = 0

It is now in Quadratic form to solve for Sin(x)

Sin(x) = { -1 +/-sqrt[1^(2) - 4(1)(-1)]} / 2(1)

Sin(x) = { -1 +/-sqrt[5[} / 2

Sin(x) = {-1 +/-2.236067978... ] / 2

Sin(x) = -3.236067978...] / 2

Sin(x) = -1.61803.... ( This is unresolved as Sine values can only range from '1' to '-1')

&

Sin(x) = 1.236067978... / 2

Sin(x) = 0.618033989...

x = Sin^(-1) [ 0.618033989...]

x = 38.17270765.... degrees.

4Sin(theta) = 2

Sin(Theta) = 2/4 = 1/2 - 0.5

Theta = Sin^(-1) [0.5]

Theta = 30 degrees.

Sin(90)= 1.000

Cos(0) = 1.000

Tan(45) = 1.000

NB The angular values repeat every 360 degrees.

Cosine (23) = Cos(23 degrees) = 0.920504853... ~ 0.9205

The value of cos 46 degrees is approximately 0.7193. Cosine is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse in a right triangle. In this case, when the angle is 46 degrees, the cosine value is calculated by dividing the length of the adjacent side by the length of the hypotenuse.

Oh honey, the sum of the first 100 odd numbers is 10,000. You see, the formula for the sum of the first n odd numbers is n^2, and since 100 is the 50th odd number, 50^2 equals 10,000. So, there you have it, darling.

Well, darling, the cosecant (csc) of a 63 degree angle is simply 1/sin(63). So, grab your calculator and divide 1 by the sine of 63 degrees to get your answer. Math doesn't have to be a drag, honey!

Well, honey, to round 10.9 to 1 decimal place, you just look at the digit right after the decimal point. If it's 5 or greater, you round up. So, 10.9 rounded to 1 decimal place is 11. Easy peasy, lemon squeezy!

Do you mean (((216^(1/2))^(1/2))^(1/2)) or [216^(1/2)] / 3

The first answer is 216^(1/8) = 1.957973....

The second one is 216^(1/2) / 3 = 4.898979486....

Oh honey, without the Order of Operations, you're just asking for a hot mess. Trying to solve complex math problems without following the correct order of operations is like trying to bake a cake without a recipe - you're gonna end up with a disaster. So, technically speaking, any math problem becomes a nightmare without those rules in place.

The cotangent function is the reciprocal of the tangent function, so cot(115 degrees) is equivalent to 1/tan(115 degrees). Since tan(115 degrees) is equivalent to -tan(65 degrees) due to the periodicity of the tangent function, cot(115 degrees) simplifies to -tan(65 degrees), which corresponds to option A.

Sine(30) = 0.5 or 1/2

Assuming the '12 inch' is the radius.

Then

C = 2 pi r

C = 2 pi 12

C = 24 pi ( Thr asnswer in terms of 'pi').

is called the 'origin' , and has the co-ordinates of (0,0)

Use Pythagoras

h^(2) = H^(2) + b^(2)

h(hypotenuse is the ladder length at 39 ft.

H is wall height which is 36 ft.

Substitute

39^(2) = 36^(2) + b^(2)

Algebraically rearrange

b^(2) = 39^(2) - 36^(2)

b^(2) = ( 39 - 36)(39 + 36)

b^(2) 3(75)

b^(2) = 225

Square root BOTH sides

b = 15 ft .

The foot of the ladder should 15feet from the base of the house wall .

tan a=1/2 tan b=1/3 find tan (a-b)

Well, darling, to find the area of a triangle with those side lengths, you can use Heron's formula. So, plug in those side lengths (a=8, b=11, c=15) into the formula, calculate the semi-perimeter, and then solve for the area. Voilà, you've got yourself the triangle's area.

To find the height of the tree, we can use trigonometry. First, we need to find the distance from Scott's eye level to the top of the tree, which is 80 ft * tan(42 degrees). This distance is the height of the tree minus Scott's eye level (4.5 ft). So, the height of the tree would be the distance + Scott's eye level, which is (80 ft * tan(42 degrees)) + 4.5 ft. Calculating this gives us the height of the tree as approximately 64.5 feet.