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Basic trigonometric functions

The basic trigonometric functions are used to solve problems involving triangles, and they are also used in many other areas of mathematics, such as calculus and physics.
  • Sine (sin): The sine of an angle is the ratio of the opposite side to the hypotenuse of a right triangle.
  • Cosine (cos): The cosine of an angle is the ratio of the adjacent side to the hypotenuse of a right triangle.
  • Tangent (tan): The tangent of an angle is the ratio of the opposite side to the adjacent side of a right triangle.
  • Cotangent (cot): The cotangent of an angle is the reciprocal of the tangent of an angle.
  • Secant (sec): The secant of an angle is the reciprocal of the cosine of an angle.
  • Cosecant (csc): The cosecant of an angle is the reciprocal of the sine of an angle.

Here are some examples of how the basic trigonometric functions can be used to solve problems involving triangles:

  • To find the missing side of a right triangle, you can use the sine, cosine, or tangent function to relate the missing side to the other two sides.
  • To find the area of a triangle, you can use the sine function to relate the area to the base and the height of the triangle.
  • To find the volume of a pyramid, you can use the sine function to relate the volume to the area of the base and the height of the pyramid.

Trigonometric identities

These identities are very useful for simplifying trigonometric expressions and solving trigonometric equations.

  • Pythagorean identity: This identity states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
    • a² + b² = c²
  • Sine addition formula: This formula gives the sine of the sum of two angles.
    • sin(α + β) = sin α cos β + cos α sin β
  • Cosine addition formula: This formula gives the cosine of the sum of two angles.
    • cos(α + β) = cos α cos β - sin α sin β
  • Tangent addition formula: This formula gives the tangent of the sum of two angles.
    • tan(α + β) = (tan α + tan β) / (1 - tan α tan β)
  • Sine subtraction formula: This formula gives the sine of the difference of two angles.
    • sin(α - β) = sin α cos β - cos α sin β
  • Cosine subtraction formula: This formula gives the cosine of the difference of two angles.
    • cos(α - β) = cos α cos β + sin α sin β
  • Tangent subtraction formula: This formula gives the tangent of the difference of two angles.
    • tan(α - β) = (tan α - tan β) / (1 + tan α tan β)

Trigonometric equations are equations that involve trigonometric functions. Some examples of trigonometric equations include:

  • sin x = 1/2
  • cos 2x = √3/2
  • tan x = √3

Trigonometric equations 

Solving trigonometric equations is the process of finding all the solutions to a trigonometric equation. There are many different methods for solving trigonometric equations, and the best method to use will depend on the specific equation. Some common methods for solving trigonometric equations include:

  • Using the unit circle
  • Using the trigonometric identities
  • Using the double angle formulas
  • Using the half angle formulas

Factoring trigonometric equations is the process of breaking a trigonometric equation down into a product of simpler trigonometric expressions. Factoring trigonometric equations can be helpful for solving trigonometric equations, and it can also be helpful for simplifying trigonometric expressions. For example, the equation sin x + cos x = 1 can be factored as sin x + cos x = (sin x + cos x)(1) = sin x cos x + 1.

Reciprocal trigonometric equations are equations that involve the reciprocals of the trigonometric functions. Reciprocal trigonometric equations can be solved using the same methods as regular trigonometric equations. For example, the equation 1/sin x = 1/cos x can be solved as sin x = cos x.

Double angle trigonometric equations are equations that involve the double angle formulas for the trigonometric functions. Double angle trigonometric equations can be solved using the double angle formulas, or they can be solved by converting them to regular trigonometric equations. For example, the equation sin 2x = 1 can be solved using the double angle formula for sin x, which is sin 2x = 2 sin x cos x.

Half angle trigonometric equations are equations that involve the half angle formulas for the trigonometric functions. Half angle trigonometric equations can be solved using the half angle formulas, or they can be solved by converting them to regular trigonometric equations. For example, the equation tan x = √3 can be solved using the half angle formula for tan x, which is tan x = 2 tan(x/2) / 1 - tan2(x/2).

Examples

Q1: A tower has a height of 50 m. It has an angle of elevation from two points on the ground level on its opposite sides are 30 degree and 60 degree respectively. Calculate the distance between two points.
(a)110.97 m
(b) 107.56 m
(c) 115.47 m
(d) 120.93 m
(e) None of the above
Ans:
c


Q2: The value of 32cot2π/4 - 8sec2π/3 + 8cos3π/6
(a) √3
(b) 2√3
(c) 3
(d) 3√3
(e) None of the above/More than one of the above.
Ans:
d

The document Overview: Trigonometry | Quantitative Aptitude for SSC CGL is a part of the SSC CGL Course Quantitative Aptitude for SSC CGL.
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FAQs on Overview: Trigonometry - Quantitative Aptitude for SSC CGL

1. What are the basic trigonometric functions?
Ans. The basic trigonometric functions are sine (sin), cosine (cos), and tangent (tan). They are used to relate the angles of a triangle to the lengths of its sides.
2. How is the sine function defined?
Ans. The sine function is defined as the ratio of the length of the side opposite to an angle in a right triangle to the length of the hypotenuse. It is represented as sin(theta) = opposite/hypotenuse.
3. What is the cosine function used for?
Ans. The cosine function is used to find the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle. It is represented as cos(theta) = adjacent/hypotenuse.
4. How can the tangent function be calculated?
Ans. The tangent function can be calculated as the ratio of the length of the opposite side to the length of the adjacent side in a right triangle. It is represented as tan(theta) = opposite/adjacent.
5. What are some practical applications of trigonometric functions?
Ans. Trigonometric functions are used in various fields such as physics, engineering, and navigation. They are used to calculate distances, angles, heights, and trajectories in real-world problems. Some examples include calculating the height of a building, determining angles of elevation and depression, and analyzing oscillations in mechanical systems.
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