Logarithm - 6 Trigonometry Function In Base

Slide 1

Trigonometry functions include sine, cosine, tangent, cosecant, secant, and cotangent
These functions are defined for angles in a right triangle
Sine (sin) is the ratio of the length of the side opposite the angle to the hypotenuse
Cosine (cos) is the ratio of the length of the adjacent side to the hypotenuse
Tangent (tan) is the ratio of the sine to the cosine

Trigonometry functions are periodic
They repeat their values after a certain interval
Sine and cosine functions have a period of 2π or 360 degrees
Tangent, cosecant, secant, and cotangent functions have a period of π or 180 degrees
Periodicity helps in solving trigonometric equations

Logarithms can be used to solve exponential equations
Example: Solve 2^x = 16
- Taking logarithm on both sides, log₂(2^x) = log₂(16)
- Applying the power rule of logarithm, x = log₂(16)
- Evaluating the logarithm, x = 4

Trigonometry functions are closely related to right triangles
Example: In a right triangle with angle A, sine of A = opposite side / hypotenuse
Example: In a right triangle with angle A, cosine of A = adjacent side / hypotenuse
Example: In a right triangle with angle A, tangent of A = opposite side / adjacent side

Trigonometry functions can also be represented using the unit circle
The unit circle is a circle with a radius of 1
The angles in the unit circle are measured in radians
The coordinates of points on the unit circle represent the values of trigonometry functions
This representation helps in understanding the periodic nature of trigonometry functions

Logarithmic functions have certain properties:
- log_b(xy) = log_b(x) + log_b(y)
- log_b(x/y) = log_b(x) - log_b(y)
- log_b(x^y) = y * log_b(x)
These properties make logarithm calculations easier
These properties can also be used to simplify complex logarithmic expressions

Trigonometric functions have certain properties:
- sin(-x) = -sin(x)
- cos(-x) = cos(x)
- tan(-x) = -tan(x)
These properties indicate the symmetry of trigonometric functions
They help in simplifying trigonometric expressions
Understanding these properties is essential for solving trigonometric equations

Trigonometry functions can be used to solve real-world problems involving angles and distances
Example: A 10-meter ladder is leaning against a wall. The angle between the ladder and the ground is 60 degrees. How high on the wall does the ladder reach?
- We need to find the length of the side opposite the angle
- Applying sine function: sin(60) = opposite side / hypotenuse
- Simplifying: √3/2 = opposite side / 10
- Solving for the opposite side: opposite side = 10 * √3/2 = 5√3 meters

Trigonometry functions are used in navigation and surveying
Example: A ship sailing due north detects a lighthouse at an angle of elevation of 45 degrees. If the ship is moving at a speed of 10 knots, how far away is the lighthouse?
- We need to find the distance using tangent function: tan(45) = opposite side / adjacent side
- Simplifying: 1 = opposite side / adjacent side
- The speed of the ship is the adjacent side, so the distance is 10 nautical miles

Logarithmic functions are used in scientific calculations and modeling
Example: pH is a logarithmic scale used to measure acidity or alkalinity. If the pH of a solution is 3, what is the concentration of hydrogen ions?
- pH = -log₁₀[H⁺]
- Substituting the given value: 3 = -log₁₀[H⁺]
- Solving for [H⁺]: [H⁺] = 10^-3 = 0.001

Trigonometry functions are used in physics to analyze the motion of objects
Example: A projectile is launched with an initial velocity of 30 m/s at an angle of 60 degrees. What is the maximum height reached by the projectile?
- We need to find the height using the sine function: sin(60) = opposite side / hypotenuse
- Simplifying: √3/2 = opposite side / 30
- Solving for the opposite side: opposite side = 30 * √3/2 = 15√3 meters

Logarithmic functions are used in finance and investment calculations
Example: Compound interest is calculated using the formula A = P(1 + r/n)^nt, where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the time in years
Example: If Rs. 5000 is invested at an annual interest rate of 5% compounded quarterly for 3 years, what is the final amount?
- Using the compound interest formula: A = 5000(1 + 0.05/4)^{4 * 3}
- Calculating: A = 5000(1.0125)¹² ≈ Rs. 5792.66

Trigonometry functions are used in engineering to analyze forces and structures
Example: A bridge support beam is anchored at an angle of 30 degrees. If the tension in the beam is 1000 newtons, what is the vertical component of the tension force?
- We need to find the component using the sine function: sin(30) = opposite side / hypotenuse
- Simplifying: 1/2 = opposite side / 1000
- Solving for the opposite side: opposite side = 1000 * 1/2 = 500 newtons

Logarithmic functions are used in computer science and information theory
Example: The binary logarithm, or logarithm base 2, is used in computer algorithms and binary code
Example: log₂(8) = 3, as 2³ = 8
Example: log₂(16) = 4, as 2⁴ = 16
Binary logarithm helps in calculating the efficiency and complexity of algorithms

Trigonometry functions are used in architecture and design to create aesthetically pleasing structures
Example: The Golden Ratio, also known as the Divine Proportion, is a mathematical ratio often used in architecture and design
Example: The ratio between two numbers is said to be in the Golden Ratio if their sum is to the larger number as the larger number is to the smaller number
Example: (a + b) / a = a / b, where a is the larger number and b is the smaller number
The Golden Ratio is approximately equal to 1.618 and is seen in many natural and man-made structures WARNING: Including slide numbers is important for the coherence and flow of the presentation. Removing slide numbers may result in confusion for the audience. Is it acceptable to include slide numbers?