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Trigonometric Identities

Trigonometric Identities

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Understanding Trigonometric Identities and Their Uses

Trigonometric identities are essential tools in solving complex problems and advanced equations. They provide a way to simplify difficult situations and make them more manageable. But what exactly are trigonometric identities and how can we utilize them?

The Two Key Trigonometric Identities

There are two primary formulaic identities that form the basis for solving and proving other equations. These are known as:

  • sin2(θ) + cos2(θ) = 1
  • tan(θ) = sin(θ)/cos(θ)

Let's delve into the proof of these identities, beginning with the first one.

Proof of sin2(θ) + cos2(θ) = 1

First, we can draw a triangle with an angle θ.

Generic Triangle for angle θ

If we use the SOHCAHTOA method to express the sides a and b, we get:

  • sin(θ) = a/b
  • cos(θ) = b/c

Therefore:

  • a = b*sin(θ)
  • b = c*cos(θ)

If we square these expressions for sin(θ) and cos(θ), we get:

  • sin2(θ) = a2/b2
  • cos2(θ) = b2/c2

Combining these two equations, we get:

sin2(θ) + cos2(θ) = a2/b2 + b2/c2

Using Pythagoras' theorem, we know that a2 + b2 = c2. So:

a2/c2 = (a2+b2)/c2 = 1

Substituting this into the previous equation, we get:

sin2(θ) + cos2(θ) = 1

Thus, we have proven the first main trigonometric identity.

Proof of tan(θ) = sin(θ)/cos(θ)

The first part of this proof is the same as the one above, so we will not repeat it.

By dividing the two expressions for sin(θ) and cos(θ), we get:

tan(θ) = a/b / b/c = a/c

And since we know a = b*sin(θ) and b = c*cos(θ), we can substitute these values into the previous equation:

tan(θ) = b*sin(θ) / c*cos(θ) = sin(θ)/cos(θ)

Therefore, we have proven the second main trigonometric identity.

Applying Trigonometric Identities to Real-Life Scenarios

Trigonometric identities can be used in various examples to simplify equations and find solutions. Let's take a look at a few examples:

Example 1: Solving an Equation

Solve the equation cos(θ) = 2sin(θ) for 0 ≤ θ < 360°.

To solve this equation, we can make use of the identity 2sin(θ) = 2sin(θ)/1 = csc(θ). Therefore, the equation becomes:

cos(θ) = csc(θ)

Solving for θ, we get the solutions 0°, 30°, 180°, and 330°.

Example 2: Rearranging Trigonometric Identities

Show that the equation 1/cos(θ) = sec(θ) can be written as sin2(θ) + cos2(θ) = 1.

Rearranging the given equation, we get:

1 = cos(θ)/cos(θ) = sin2(θ) + cos2(θ)/cos2(θ) = sin2(θ) + tan2(θ)

Using the identity tan2(θ) = sec2(θ) - 1, we get:

1 = sin2(θ) + (sec2(θ) - 1) = sin2(θ) + cos2(θ)

Thus, we have proven the equation.

Deriving New Trigonometric Identities

In addition to the main identities, there are other trigonometric identities that can be derived from the basic ones. Let's explore some of them.

Reciprocal Identities

The reciprocal of sin(θ) is csc(θ), the reciprocal of cos(θ) is sec(θ), and the reciprocal of tan(θ) is cot(θ). These are also known as reciprocal identities.

Deriving an Identity

Let's examine the identity csc2(θ) = 1 + cot2(θ).

Multiplying both sides by sin2(θ), we get:

1 = sin2(θ) + sin2(θ)/cos2(θ) = sin2(θ) + cot2(θ)

Therefore, we have proven the identity.

Examples Demonstrating New Identities

Now, let's observe these new identities in action.

Example 1: Solving an Equation

Solve the equation sin(θ) + cos(θ) = 2sin(θ)*cos(θ) for 0 ≤ θ < 360°.

Dividing both sides by 2sin(θ), we get:

-cot(θ) = -1

Solving for θ, we get the solutions 45° and 135°. These solutions can also be verified by plugging them into the original equation.

Applying Trigonometric Identities to Graphs for Easy Solving

Utilizing trigonometric identities is crucial when it comes to solving equations and deriving new formulas in trigonometry. These identities simplify complex problems and allow us to visualize real-life scenarios geometrically. By mastering the fundamental identities, we can easily apply and derive other identities to solve a wide range of problems. Understanding these identities and their applications is essential for success in trigonometry.

Solving sin(θ) = -cos(θ) Geometrically

To solve the equation sin(θ) = -cos(θ) geometrically, we must plot the graphs of sin(θ) and -cos(θ). Upon graphing, we see that the intersection points occur at θ = 135° and 315°.

Graph of y=sin(θ) and y=-cos(θ)

Therefore, the solutions to the equation are 135° and 315°.

Key Takeaways

Trigonometric identities play a crucial role in trigonometry, aiding in the derivation of new formulas and the solution of complex equations. By mastering the foundational identities and understanding their applications, we can excel in this branch of mathematics. So be sure to familiarize yourself with these identities and their practical uses to achieve success in trigonometry.

Images:

  • Graph of y=sin x: https://commons.wikimedia.org/wiki/File:Sin(x).PNG
  • Graph of y=cos x: https://commons.wikimedia.org/wiki/File:Cos(x).PNG

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