Trigonometry

"Non sine palmere palman" is pronounced as "non see-nay pal-veh-ray pal-man." The emphasis is typically placed on the syllables "pal" and "man." It is a Latin phrase, and pronunciation may vary slightly based on regional accents or classical versus ecclesiastical Latin.

To find the linear speed of the flywheel, first calculate its circumference using the formula (C = \pi \times d), where (d) is the diameter. For a 12 ft diameter, the circumference is approximately (37.7) ft. At 80 revolutions per minute, the linear speed is (80 \times 37.7 \approx 3016) ft/min. Thus, the flywheel travels about 3016 feet per minute.

Three methods to resolve a system of forces include the graphical method, where forces are represented as vectors on a diagram, and their resultant is determined visually; the analytical method, which involves using mathematical equations to sum the forces in different directions; and the method of components, where each force is broken down into its horizontal and vertical components, allowing for easier calculation of the resultant force. Each method provides a systematic approach to understanding and analyzing the effects of multiple forces acting on an object.

Plane trigonometry can be used in making a dress pattern by applying geometric principles to accurately calculate angles and lengths needed for various garment pieces. For instance, trigonometric functions can help determine the slope of armholes or necklines, ensuring proper fit and proportions. Additionally, it can assist in creating curved shapes by calculating the necessary radius for circular or curved hems. By using trigonometry, designers can create patterns that fit well and maintain aesthetic appeal.

The word "trigonometry" belongs to the category of mathematics. It is a branch that deals with the relationships between the angles and sides of triangles, particularly right triangles. Trigonometry is essential in various fields, including physics, engineering, and astronomy, as it helps in solving problems involving angles and distances.

The principles of trigonometry revolve around the relationships between the angles and sides of triangles, particularly right triangles. Key concepts include the sine, cosine, and tangent functions, which relate the angles to the ratios of the lengths of the sides. Additionally, the Pythagorean theorem establishes a fundamental relationship between the sides of a right triangle. Trigonometry is also essential in studying periodic phenomena, such as waves and oscillations, through its functions and identities.

One!!!!

It is a radial line, from the centre point of the base/straight line to the midpoint of the circumference.

Sine is applied alomg the x-axis

Sinh is applied along the y-axis.

Similarly 'Cosh', and 'Tanh'.

Pronounced

Sinh = Shin

Cassh = Cosh

Tanh = Than

The Sine wave is applied to infinity along the x-axis, with a MAX/MIN AT y = '1/-1.

Correspondingly

Sinh wave is applied to infinity along the y-axis, with a max/min at x = 1/-1.

To calculate shaft taper, measure the diameter at two points along the length of the shaft. Subtract the smaller diameter from the larger diameter to find the taper amount. Then, divide this difference by the length of the taper section to determine the taper per unit length. This result is often expressed as a ratio or in degrees, depending on the application.

The cosine of 70 degrees is approximately 0.342. This value can be found using a scientific calculator or trigonometric tables. In the context of a right triangle, it represents the ratio of the length of the adjacent side to the hypotenuse for an angle of 70 degrees.

Georg Joachim Iserin, also known as Georg Tullius, made significant contributions to trigonometry by introducing the concept of the "sine" function in his work "Trigonometry" published in 1543. He was one of the first to use the sine table, which simplified the calculations of angles and distances. His work laid the groundwork for more advanced trigonometric concepts and calculations, influencing later mathematicians and astronomers in their studies. Additionally, Iserin's approach helped to transition trigonometry from a purely geometric discipline to a more analytical one.

To find the recoil velocity of the Earth when a 5 kg bowling ball is projected upward with a velocity of 2.0 meters per second, we can use the principle of conservation of momentum. Initially, the total momentum is zero, so the momentum gained by the bowling ball must equal the momentum lost by the Earth. The momentum of the bowling ball is ( p = mv = 5 , \text{kg} \times 2 , \text{m/s} = 10 , \text{kg m/s} ). Since the mass of the Earth is approximately ( 5.97 \times 10^{24} , \text{kg} ), the recoil velocity of the Earth can be calculated as ( v_{Earth} = -\frac{p_{ball}}{m_{Earth}} = -\frac{10}{5.97 \times 10^{24}} \approx -1.67 \times 10^{-24} , \text{m/s} ), indicating an extremely small downward velocity.

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.