# Trigonometry Formulas, Examples and Solutions!

Trigonometry is the study of relationships between angles and sides of triangles and is a fundamental branch of mathematics. However, Trigonometry Formulas come into use whilst solving trigonometry problems. Furthermore, these Trigonometry Formulas help with the solutions. Read on to learn more about these important Trigonometry Formulas along with their Examples and Solutions.

## Basic Trigonometric Ratio Formulas

The Basic Trigonometric Ratios are sine (sin), cosine (cos), and tangent (tan). Moreover, these Ratios are defined as follows:

• Sine (sin): The ratio of the opposite side to the hypotenuse of a right-angled triangle.
• For Example: In a right-angled triangle with an angle of 30°, if the hypotenuse is 10 units, the opposite side is 5 units.
• Therefore, sin(30°) = 0.5.
• Cosine (cos): The ratio of the adjacent side to the hypotenuse of a right-angled triangle.
• For Example: In the same triangle, the adjacent side is 8.66 units.
• Therefore, cos(30°) = 0.866.
• Tangent (tan): The ratio of the opposite side to the adjacent side of a right-angled triangle.
• For Example: In the same triangle, the tangent of 30° is 0.577, as the opposite side is 5 units and the adjacent side is 8.66 units.

## Reciprocal Identities

1. Cosecant (csc): The reciprocal of sine, csc(x) = 1/sin(x).
1. Secant (sec): The reciprocal of cosine, sec(x) = 1/cos(x).
1. Cotangent (cot): The reciprocal of tangent, cot(x) = 1/tan(x).

For Example:

If sin(x) = 0.5,

then csc(x) = 1/0.5 = 2.

Also Read: Algebraic Identities: Examples and Chart

## Trigonometric Ratio Table

Here is the Trigonometric Ratio Table that provides the values of sine, cosine, and tangent for common angles:

For Example: If the angle is 45°, the sine is 0.707, the cosine is 0.707, and the tangent is 1.

## Periodic Identities

Moreover, the Periodic Identities describe the repeating patterns of Trigonometric functions:

1. sin(x + 2π) = sin(x)
1. cos(x + 2π) = cos(x)
1. tan(x + π) = -tan(x)

For Example:

If sin(x) = 0.5,

then sin(x + 2π) = 0.5.

Also Read: Order of Operations and PEMDAS Rule

## Co-Function Identities

The Co-Function Identities relate the Trigonometric functions of Complementary angles (angles that add up to 90°):

1. sin(x) = cos(90° – x)
1. cos(x) = sin(90° – x)
1. tan(x) = cot(90° – x)

For Example:

If sin(30°) = 0.5,

then cos(60°) = 0.5.

## Sum and Difference Identities

The Sum and Difference Identities describe the relationships between Trigonometric functions of the sum or difference of two angles:

1. sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
1. sin(A – B) = sin(A)cos(B) – cos(A)sin(B)
1. cos(A + B) = cos(A)cos(B) – sin(A)sin(B)
1. cos(A – B) = cos(A)cos(B) + sin(A)sin(B)

For Example:

If sin(30°) = 0.5 and cos(45°) = 0.707,

then sin(30° + 45°) = 0.5 ✕ 0.707 + 0.866 ✕ 0.707 = 0.866.

## Half, Double and Triple Identities

These Identities relate the Trigonometric functions of an angle to the functions of Half, Double, or Triple that angle:

1. sin(2x) = 2sin(x)cos(x)
1. cos(2x) = cos²(x) – sin²(x)
1. tan(2x) = 2tan(x) / (1 – tan²(x))

For Example:

If sin(30°) = 0.5,

then sin(60°) = 2 ✕ 0.5 ✕ 0.866 = 0.866.

## Sum to Product Identities

The Sum to Product Identities allows the conversion of sums and differences of Trigonometric functions into products:

1. sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)
1. sin(A) – sin(B) = 2cos((A+B)/2)sin((A-B)/2)
1. cos(A) + cos(B) = 2cos((A+B)/2)cos((A-B)/2)
1. cos(A) – cos(B) = -2sin((A+B)/2)sin((A-B)/2)

For Example:

If sin(30°) = 0.5 and sin(45°) = 0.707,

then sin(30°) + sin(45°) = 2 ✕ sin(75°/2) ✕ cos(15°/2) = 0.866 ✕ 0.966 = 0.837.

Also Read: Profit and Loss Formula Questions

## Inverse Trigonometry Formulas

The inverse trigonometric functions, denoted as sin^-1, cos^-1, and tan^-1, allow us to find the angle given the value of the trigonometric function:

1. sin^-1(x) = the angle whose sine is x
2. cos^-1(x) = the angle whose cosine is x
3. tan^-1(x) = the angle whose tangent is x

For Example:

If sin(x) = 0.5,

then x = sin^-1(0.5) = 30°.

Also Read: 20 Most Famous Indian Mathematicians

## Sine Law

The Sine Law states that the ratio of the length of a side to the sine of the opposite angle is constant for all triangles:

sin(A)/a = sin(B)/b = sin(C)/c

For Example:

In a triangle with sides a = 5, b = 6, and c = 7, and angles A, B, and C, we can use the sine law to find the unknown angle A:

sin(A)/5 = sin(B)/6

sin(A) = (5/6)sin(B)

## Cosine Law

The Cosine Law relates the lengths of the sides of a triangle to the cosine of one of the angles:

c^2 = a^2 + b^2 – 2ab cos(C)

For Example:

In a triangle with sides a = 5, b = 6, and c = 7, we can use the cosine law to find the angle C:

7^2 = 5^2 + 6^2 – 2(5)(6)cos(C)

cos(C) = (5^2 + 6^2 – 7^2) / (2 * 5 * 6) = 0.5