JEE Mains

Maximum and minimum values of trigonometric functions:

• The maximum value of sin x is 1, which occurs when x = (\frac{\pi}{2})+ 2n(\pi).
• The minimum value of sin x is -1, which occurs when x = (\frac{3\pi}{2})+ 2n(\pi).
• The maximum value of cos x is 1, which occurs when x = 2n(\pi).
• The minimum value of cos x is -1, which occurs when x = (\pi)+ 2n(\pi).
• The maximum value of tan x is (\infty), which occurs when x = (\frac{\pi}{2})+ n(\pi).
• The minimum value of tan x is (-\infty), which occurs when x = (\frac{3\pi}{2})+ n(\pi).

Range of trigonometric functions:

• The range of sin x is [-1, 1].
• The range of cos x is [-1, 1].
• The range of tan x is (-\infty\ <\tan x<\infty).

Periodicity of trigonometric functions:

• The period of sin x is 2(\pi).
• The period of cos x is 2(\pi).
• The period of tan x is (\pi).

Inverse trigonometric functions:

• The inverse of sin x is arcsin x, which is defined for -1 (\le x\le)1.
• The inverse of cos x is arccos x, which is defined for -1 (\le x\le)1.
• The inverse of tan x is arctan x, which is defined for all real numbers.

Properties of trigonometric functions:

• sin(-x) = -sin x
• cos(-x) = cos x
• tan(-x) = -tan x
• sin (x+y) = sin x cos y + cos x sin y
• cos (x+y) = cos x cos y - sin x sin y
• tan (x+y) = (\frac{\tan x + \tan y}{1 - \tan x \tan y})

Graphs of trigonometric functions:

• The graph of sin x is a sine curve.
• The graph of cos x is a cosine curve.
• The graph of tan x is a tangent curve.

Applications of trigonometric functions to real-world problems:

• Trigonometry is used in navigation to find the direction and distance to a destination.
• Trigonometry is used in surveying to measure the distance between two points.
• Trigonometry is used in astronomy to calculate the positions and distances of stars and planets.
• Trigonometry is used in engineering to design bridges, buildings, and other structures.

Trigonometric identities:

• Pythagorean identity: sin^2 x + cos^2 x = 1
• Cofunction identities: sin x = cos ((\frac{\pi}{2} - x)), cos x = sin ((\frac{\pi}{2} - x)), and tan x = cot ((\frac{\pi}{2} - x))
• Sum and difference identities: sin (x+y) = sin x cos y + cos x sin y, cos (x+y) = cos x cos y - sin x sin y, tan (x+y) = (\frac{\tan x + \tan y}{1 - \tan x \tan y}), sin (x-y) = sin x cos y - cos x sin y, cos (x-y) = cos x cos y + sin x sin y, and tan (x-y) = (\frac{\tan x - \tan y}{1 + \tan x \tan y}).
• Double angle identities: sin 2x = 2 sin x cos x, cos 2x = cos^2 x - sin^2 x, and tan 2x = (\frac{2 \tan x}{1 - \tan^2 x}).
• Half angle identities: sin (\frac{x}{2}) = (\pm\sqrt{(-1+\cos x)}\2) cos (\frac{x}{2}) = (\pm\sqrt{(-1+\cos x)}\2), tan (\frac{x}{2}) = (\frac{sin x}{1+cos x})

Solution of trigonometric equations:

• Trigonometric equations can be solved using a variety of methods, including:
  • Graphical methods
  • Algebraic methods
  • Numerical methods

CBSE Board Exams

Basic trigonometric ratios (sin, cos, tan, cosec, sec, cot):

• The six trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent.
• Sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle.
• Cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
• Tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side in a right triangle.
• Cosecant of an angle is the reciprocal of the sine of that angle.
• Secant of an angle is the reciprocal of the cosine of that angle.
• Cotangent of an angle is the reciprocal of the tangent of that angle. Complementary angles
• Two angles are complementary if they add up to 90 degrees.
• For example, 30 degrees and 60 degrees are complementary angles.

Supplementary angles

• Two angles are supplementary if they add up to 180 degrees.
• For example, 45 degrees and 135 degrees are supplementary angles.

Trigonometric identities:

• The Pythagorean 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.
• The sum and difference identities for sine and cosine are used to simplify trigonometric expressions.
• The identities for double and half angles are useful for finding the values of trigonometric functions for angles that are multiples or fractions of other angles. Solution of simple trigonometric equations
• Simple trigonometric equations can be solved using a variety of methods, including:

• Substitution
• Factorisation
• Using trigonometric identities

Applications of trigonometry to real-world problems Trigonometry is used in a variety of real-world applications, including: -测量高度和距离、

• Navigation, surveying,
• Engineering,
• Astronomy,
• Cartography,
• Robotics.