Trigonometry Fundamentals and Applications

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Questions and Answers

What theorem relates the lengths of the sides in a right triangle?

The Sine Rule

The Pythagorean Theorem (correct)

The Area Formula

The Cosine Rule

Which trigonometric function represents the ratio of the opposite side to the hypotenuse in a right triangle?

Sine (correct)

Secant

Tangent

Cosine

If the angle A in a right triangle is 30 degrees, what is tan(A)?

$\sqrt{3}/3$

$\sqrt{3}$

$1/\sqrt{3}$ (correct)

$1/3$

What is the reciprocal function of the cosine function?

Secant Signup and view all the answers

Which trigonometric function is used to find the ratio of the adjacent side to the hypotenuse in a right triangle?

Cosine Signup and view all the answers

If the angle B in a right triangle is 60 degrees, what is cot(B)?

$1/\sqrt{3}$ Signup and view all the answers

What fundamental experiments were performed by scientists in the 18th and 19th century?

Experiments to measure the substances used and formed during chemical changes Signup and view all the answers

Who were the scientists that studied the structure of substances and the atom, leading to the discovery of the laws of chemical combination?

Dalton, Thomson, and Rutherford Signup and view all the answers

What is the significance of Dalton’s atomic theory in writing molecular formulae of compounds?

It provided the basis for writing molecular formulae Signup and view all the answers

What apparatus is recommended for verifying the laws of chemical combination?

Conical flask, test tubes, balance, etc. Signup and view all the answers

What chemicals are used in the activity involving calcium chloride and sodium sulphate solutions?

Calcium chloride (CaCl2) and sodium sulphate (Na2SO4) Signup and view all the answers

What should be done after inserting the test tube with sodium sulphate solution into the conical flask?

Seal the conical flask with an airtight rubber cork Signup and view all the answers

What is the purpose of weighing the conical flask before and after pouring the solution?

To measure the change in mass and verify the laws of chemical combination Signup and view all the answers

What happens when 56 g of calcium oxide is mixed with 18 g of water?

A reaction occurs Signup and view all the answers

What inference can be drawn from observing the reaction between calcium oxide and water?

A new substance is formed Signup and view all the answers

How do the experiments with calcium chloride, sodium sulphate, calcium oxide, and water relate to the laws of chemical combination?

They help in verifying the laws through practical demonstrations Signup and view all the answers

Study Notes

Trigonometry

Solving Triangles

To solve triangles, we use the Pythagorean Theorem. It states that the square of the length of the hypotenuse is equal to the sum of squares of the lengths of the other two sides. Mathematically: h² = l² + b².

Trigonometric Equations

Trigonometric equations are used to solve problems involving angles and their relationships with side lengths in triangles. There are six basic trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These functions relate the sides of a right triangle to one another, based on the angle being measured. For example, sin(A) is the ratio of the length of the side opposite an acute angle A to the hypotenuse. Similarly, cos(A) is the ratio of the length of the adjacent side to the hypotenuse. tan(A) is the ratio of the length of the side opposite to A over the length of the side adjacent to A, and so on.

Trigonometric Functions

The six fundamental trigonometric functions are used to calculate the values of other trigonometric functions for different angles. They are defined as follows:

Sine: sin(θ) = opp / hyp
Cosine: cos(θ) = adj / hyp
Tangent: tan(θ) = opp / adj
Cosecant: csc(θ) = 1 / sin(θ)
Secant: sec(θ) = 1 / cos(θ)
Cotangent: cot(θ) = 1 / tan(θ)

These functions have various applications in different fields such as astronomy, mapmaking, surveying, and artillery range finding.

Surface Area and Volume

Using trigonometry, we can find both surface area and volume of various shapes:

Surface Area of a Sphere: The formula for calculating the surface area of a sphere is 4πr^2, where r is the radius of the sphere.
Volume of a Cylinder or Prism: The formula for calculating the volume of a cylinder or prism is V = l * h, where l represents the base area and h represents the height of the cylinder or prism.

Circle Coordinate Geometry

In coordinate geometry, circles play an important role. The equation of a circle centered at (a, b) with radius r is given by (x - a)^2 + (y - b)^2 = r^2. This equation describes the relationship between points on the circle and their coordinates in a two-dimensional plane.

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Description

Learn about solving triangles using the Pythagorean Theorem, trigonometric equations, functions, surface area and volume calculations, and circle coordinate geometry. Understand how trigonometry is applied in various fields like astronomy, mapmaking, surveying, and more.