Important Formulas: Trigonometry

# Important Formulas: Trigonometry | SSC CGL Tier 2 - Study Material, Online Tests, Previous Year PDF Download

## Trigonometry Formulas

When we learn about trigonometric formulas, we consider them for right-angled triangles only. In a right-angled triangle, we have 3 sides namely – Hypotenuse, Opposite side (Perpendicular), and Adjacent side (Base). The longest side is known as the hypotenuse, the side opposite to the angle is perpendicular and the side where both hypotenuse and opposite side rests is the adjacent side.

### Basic Trigonometric Function Formulas

There are basically 6 ratios used for finding the elements in Trigonometry. They are called trigonometric functions. The six trigonometric functions are sine, cosine, secant, cosecant, tangent and cotangent.

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived:

• sin θ = Opposite Side/Hypotenuse
• cos θ = Adjacent Side/Hypotenuse
• tan θ = Opposite Side/Adjacent Side
• sec θ = Hypotenuse/Adjacent Side
• cosec θ = Hypotenuse/Opposite Side
• cot θ = Adjacent Side/Opposite Side

### Reciprocal Identities

The Reciprocal Identities are given as:

• cosec θ = 1/sin θ
• sec θ = 1/cos θ
• cot θ = 1/tan θ
• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ

All these are taken from a right-angled triangle. When the height and base side of the right triangle are known, we can find out the sine, cosine, tangent, secant, cosecant, and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.

### Trigonometry Table

Below is the table for trigonometry formulas for angles that are commonly used for solving problems.

These formulas are used to shift the angles by π/2, π, 2π, etc. They are also called co-function identities.

• sin (π/2 – A) = cos A & cos (π/2 – A) = sin A
• sin (π/2 + A) = cos A & cos (π/2 + A) = – sin A
• sin (3π/2 – A)  = – cos A & cos (3π/2 – A)  = – sin A
• sin (3π/2 + A) = – cos A & cos (3π/2 + A) = sin A
• sin (π – A) = sin A &  cos (π – A) = – cos A
• sin (π + A) = – sin A & cos (π + A) = – cos A
• sin (2π – A) = – sin A & cos (2π – A) = cos A
• sin (2π + A) = sin A & cos (2π + A) = cos A

All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identities. tan 45° = tan 225° but this is true for cos 45° and cos 225°. Refer to the above trigonometry table to verify the values.

### Cofunction Identities (in Degrees)

The co-function or periodic identities can also be represented in degrees as:

• sin(90°−x) = cos x
• cos(90°−x) = sin x
• tan(90°−x) = cot x
• cot(90°−x) = tan x
• sec(90°−x) = cosec x
• cosec(90°−x) = sec x

### Sum & Difference Identities

• sin(x+y) = sin(x)cos(y)+cos(x)sin(y)
• cos(x+y) = cos(x)cos(y)–sin(x)sin(y)
• sin(x–y) = sin(x)cos(y)–cos(x)sin(y)
• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)

### Triple Angle Identities

• Sin 3x = 3sin x – 4sin3x
• Cos 3x = 4cos3x-3cos x

Also,

### Inverse Trigonometry Formulas

• sin-1 (–x) = – sin-1 x
• cos-1 (–x) = π – cos-1 x
• tan-1 (–x) = – tan-1 x
• cosec-1 (–x) = – cosec-1 x
• sec-1 (–x) = π – sec-1 x
• cot-1 (–x) = π – cot-1 x

### What is Sin 3x Formula?

Sin 3x is the sine of three times of an angle in a right-angled triangle, which is expressed as:
Sin 3x = 3sin x – 4sin3x

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## SSC CGL Tier 2 - Study Material, Online Tests, Previous Year

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## FAQs on Important Formulas: Trigonometry - SSC CGL Tier 2 - Study Material, Online Tests, Previous Year

 1. What are the basic trigonometric ratios?
Ans. The basic trigonometric ratios are sine, cosine, and tangent. Sine (sin) is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse. Cosine (cos) is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. Tangent (tan) is defined as the ratio of the length of the side opposite to the angle to the length of the adjacent side.
 2. How do I find the values of trigonometric ratios for special angles?
Ans. For special angles such as 0°, 30°, 45°, 60°, and 90°, the values of trigonometric ratios can be easily determined. For example, for 30°, sin(30°) = 1/2, cos(30°) = √3/2, and tan(30°) = 1/√3. Similarly, for 45°, sin(45°) = cos(45°) = 1/√2, and tan(45°) = 1. These values can be memorized for quick calculations.
 3. How can I find the values of trigonometric ratios for any angle?
Ans. To find the values of trigonometric ratios for any angle, you can use a scientific calculator or refer to trigonometric tables. Simply enter the angle in degrees and use the sine, cosine, or tangent function to obtain the corresponding ratio. Alternatively, you can use the unit circle to determine the ratios for various angles.
 4. What is the Pythagorean identity in trigonometry?
Ans. The Pythagorean identity in trigonometry is a fundamental relationship between the sine and cosine functions. It states that sin^2(theta) + cos^2(theta) = 1, where theta is an angle. This identity is derived from the Pythagorean theorem in geometry and holds true for all angles.
 5. How are trigonometric ratios used in real-life applications?
Ans. Trigonometric ratios have various real-life applications. They are used in navigation to determine distances and angles between objects, in architecture and engineering for designing and constructing buildings and structures, in physics for analyzing oscillatory motion and waves, in astronomy for studying celestial bodies and their movements, and in many other fields involving calculations and measurements involving angles and triangles.

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