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Trig Identities Cheat Sheet

May 14, 2024
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Trigonometric identities are a collection of formulas used to simplify different complex trigonometric equations. Trigonometry explores the relationships between the sides and angles of a triangle. These relationships are quantified in the form of six ratios called trigonometric ratios: sin, cos, tangent, cotangent, secant, and cosecant. 

In a more general manner, the study is also about the angles corresponding to the sides of a triangle. The study of a triangle, solving a triangle, and applying an understanding of a triangle to physical tasks involving height and the distance between two points comprise the study area. Moreover, it gives a way of solving trigonometric equations.

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This article will explore the trig identities cheat sheet just like the calculus trig identities cheat sheet, etc.

What are Trigonometric Identities?

An expression involving trigonometric ratios of an angle is said to be a Trigonometric Identity if the expression is true for all values of the angle. These are applicable whenever trigonometric functions appear in expressions or equations. The six basic trigonometric ratios are sine, cosine, tangent, cosecant, secant, and cotangent. These trigonometric ratios are understood in terms of the sides of the right triangle, like the adjacent side, opposite side, and hypotenuse side.

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List of Trigonometric Identities

The study of trigonometry consists of many identities that are all about trigonometric ratios. These identities are applied to a wide range of complex situations that extend the academic domain and real life. Now, we will learn all the simple and complex trigonometric identities. 

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Reciprocal Trigonometric Identities

In all trigonometric ratios, there is a reciprocal relation between a pair of ratios, which is given as follows: 

  • csc(θ) =1/sin(θ)
  • sec(θ) = 1/cos(θ)
  • cot(θ) =1/tan(θ)

Pythagorean Trigonometric Identities

Pythagorean trigonometric identities are based on the Right-Triangle theorem or Pythagoras theorem and are as follows:

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  • sin²(θ) + cos²(θ) = 1
  • tan²(θ) + 1 = sec²(θ)
  • 1 + cot²(θ) = csc²(θ)

Sum and Difference Identities

Trigonometric identities for sum and difference of angle that come into these formulas, sin(A+B), cos(A-B), tan(A+B), etc.

  • sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β)
  • cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β)
  • tan(α ± β) =(tan(α) ± tan(β))/(1 ∓ tan(α)tan(β))

Double Angle Identities

By referring to the trigonometric identities of the sum of angles, an alternative identity known as the Double angle Identity can be derived. This way, we can verify the relationships with the same angle identities A = B. For example,

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We know, sin (A+B) = sin A cos B + cos A sin B

Substitute A = B = θ on both sides here, and we get:

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sin (θ + θ) = sinθ cosθ + cosθ sinθ

sin 2θ = 2 sinθ cosθ

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Similarly,

cos 2θ = cos2θ – sin 2θ = 2 cos 2 θ – 1 = 1 – sin 2 θ

tan 2θ = (2tanθ)/(1 – tan2θ)

  • sin(2θ) = 2sin(θ)cos(θ)
  • cos(2θ) = cos²(θ) – sin²(θ)
  • tan(2θ) =2tan(θ)/(1 – tan²(θ))

Quotient Identities

Tan and cot are defined as the ratio of sin and cos, which is given by the following identities:

  • tan(θ) =sin(θ)/cos(θ)
  • cot(θ) = cos(θ)/sin(θ)

Even-Odd Identities

In trigonometry, an angle measured in the clockwise direction is measured in negative parity, and all trigonometric ratios defined for negative parity of angle are defined as follows:

  • sin (-θ) = -sin θ
  • cos (-θ) = cos θ
  • tan (-θ) = -tan θ
  • cot (-θ) = -cot θ
  • sec (-θ) = sec θ
  • cosec (-θ) = -cosec θ

Complementary Angles Identities

Complimentary angles are pairs of angles with a sum of 90°. Now, the trigonometric identities for complementary angles are as follows:

  • sin (90° – θ) = cos θ
  • cos (90° – θ) = sin θ
  • tan (90° – θ) = cot θ
  • cot (90° – θ) = tan θ
  • sec (90° – θ) = cosec θ
  • cosec (90° – θ) = sec θ

Supplementary Angles Identities

Supplementary angles are pairs of angles that equal 180 degrees. Now, the trigonometric identities for supplementary angles are:

  • sin (180°- θ) = sinθ
  • cos (180°- θ) = -cos θ
  • cosec (180°- θ) = cosec θ
  • sec (180°- θ)= -sec θ
  • tan (180°- θ) = -tan θ
  • cot (180°- θ) = -cot θ

Periodicity of Trigonometric Function

The trigonometric functions like sin, cos, tan, cot, sec, and cosec are periodic functions that have different frequencies. The following explanation of the period for the trigonometric ratios is given:

sin (n × 360° + θ) = sin θ

sin (2nπ + θ) = sin θ

cos (n × 360° + θ) = cos θ

cos (2nπ + θ) = cos θ

tan (n × 180° + θ) = tan θ

tan (nπ + θ) = tan θ

cosec (n × 360° + θ) = cosec θ

cosec (2nπ + θ) = cosec θ

sec (n × 360° + θ) = sec θ

sec (2nπ + θ) = sec θ

cot (n × 180° + θ) = cot θ

cot (nπ + θ) = cot θ

Where,n ∈ Z, (Z = set of all integers)

Note: sin, cos, cosec, and sec have a period of 360° or 2π radians, and for tan and cot, the period is 180° or radians.

Half-Angle Formulas

Utilizing the double-angle formulas, we can calculate half-angle formulas. To get the half-angle formula, the first step is to replace θ with θ/2.

  • sin(θ/2) = ±√((1 – cos(θ)/2)
  • cos(θ/2) = ±√((1 + cos(θ)/2)
  • tan(θ/2) = ±√((1 – cos(θ)/(1 + cos(θ))

Product-Sum Identities

The sum of two trigonometric ratios can be expressed as the product of the two trigonometric ratios as follows.

1. Sine Addition Formula:

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

2. Cosine Addition Formula:

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

3. Tangent Addition Formula:

tan(α + β) = (tan(α) + tan(β) / 1 – tan(α)tan(β))

Products Identities

Product Identities are formed when we add two of the sum and difference of angle identities and are as follows:

1. Sine of a Sum:

sin(α)sin(β) = 1/2[cos(α – β) – cos(α + β)]

2. Sine of a Difference:

sin(α)cos(β) = 1/2[sin(α + β) + sin(α – β)]

3. Cosine of a Sum:

cos(α)cos(β) =  1/2[cos(α – β) + cos(α + β)]

4. Cosine of a Difference:

cos(α)sin(β) =  1/2[sin(α + β) – sin(α – β)]

Triple Angle Formulas

In addition to the double and half angle formulas, there are identities for the trigonometric ratios defined for the triple angles. These identities are as follows:

1. Sine of Triple Angle:

sin(3θ) = 3sin(θ) – 4sin³(θ)

2. Cosine of Triple Angle:

cos(3θ) = 4cos³(θ) – 3cos(θ)

3. Tangent of Triple Angle:

tan(3θ) = 3tan(θ) – tan³(θ) / 1 – 3tan²(θ)

Conclusion

Knowing the trigonometry rules is essential to performing well in math and science subjects. Through the use of an all-in-one cheat sheet, for example, the one produced by Turito, learners are able to approach trigonometry with confidence and ease. Learn more from the Turito website through various tools like tutorials and individual sessions to gain confidence and improve your grades.

FAQs 

Why do I need to memorize trigonometric identities?

Understanding trigonometric identities helps solve math problems faster and easier. They are like shortcuts that make trigonometry simpler.

How can I remember all the trig identities?

Tricks like acronyms or memorizing consistently can be effective. The more you practice, the more you can remember them.

How do I know which trig identity to use in a given problem?

Look at the problem and determine which identity best fits it. As you practice, you will become better at choosing the right one. 

Trig Identities Cheat Sheet

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