Trigonometric identities are equations that include trigonometric functions, and are correct for each value of the occurrence variable defined on both sides of the equation. Geometrically, these are identities involving certain functions of one or more angles.
The trigonometric identity is always the correct trigonometric equation. They are commonly used to solve trigonometric and geometric problems and to understand various mathematical properties. Knowing the identity of key triggers can help you remember and understand important mathematical principles and solve many mathematical problems.
Background of Trigonometric functions:
The term trigonometry is a Latin derivative of the 16th century, derived from the Greek trigōnon and Metron. Although this field appeared in Greece in the third century BC, some of the most important contributions (such as sine functions) came from India in the fifth century. With the loss of early trigonometry in ancient Greece, it is not clear whether Indian scholars developed independently or under the influence of Greece. According to Victor Katz in “History of Mathematics (3rd Edition)” (Pearson, 2008), the development of trigonometry was mainly based on the needs of Greek astronomers.
6 Basic Trigonometric Functions
There are basic 6 Trigonometric ratios which are sine, consine, tangent, cosecant, secant and cotangent.They can be expressed in terms of sides of right triangle for angle θ
Sine.
Sine is also called sin
Sine is defined as when an opposite side of a right angle triangle is divided to the hypotenuse side of a right angle triangle the answer in result will be known as sine
Cosine:
Cosine is also known cos
Cosine is defined as when the adjacent side of a right angle triangle is divided to the hypotenuse of a right angle triangle is called cosine
Tangent:
Tangent is also known as Tan
Tangent is defined as when opposite side of a right angle triangle is divided to the adjacent side of a right angle triangle the answer in result will be known as Tangent
All functions are show below
| Sine Function: | sin(θ) = Opposite / Hypotenuse |
| Cosine Function: | cos(θ) = Adjacent / Hypotenuse |
| Tangent Function: | tan(θ) = Opposite / Adjacent |
Guidelines for verifying Trigonometric Identities:
Below are the some guidelines to verify trigonometric identities.
- Check whether the statement is false.
This is easily done on a graphing calculator. Graph both sides of the identity and check to
See if you get the same picture.
- Only manipulate one side of the proposed identity until it becomes the other side of the identity.
Typically the more complicated side is the best place to start. That side will give you more
To work with.
- DO NOT treat the identity like an equation.
This assumes that the identity is true, which is the thing that you are trying to prove.
Relationships between Trigonometric Functions
- Expressing the sine in terms of cosine
sinα=±√1−cos2α
Note:The sign in front of the radical on the right side depends on the quadrant in which the angle lies. The sign and value of a trigonometric function on the left side must coincide with the sign and value in the right side. This rule also applies to other formulas given below. - Expressing the sine in terms of tangent
sinα=tanα±√1+tan2α - Expressing the sine in terms of cotangent
sinα=1±√1+cot2α - Expressing the cosine in terms of sine
cosα=±√1−sin2α - Expressing the cosine in terms of tangent
cosα=1±√1+tan2α
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