### Shortcut Methods

**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.