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Equivalent Representations of Trigonometric Functions

Lesson ~12 min read 8 MCQs

In simple terms: In simple terms, this topic is about rewriting trigonometric expressions in different but equivalent ways—like using synonyms—to make solving complex problems much easier.

Why this matters

Imagine you're a programmer designing the animation for a character in a new video game. You've coded a cool jumping arc using a sine function. But now, the lead designer wants the character to do a flip at the peak of the jump. That flip is a different motion, probably described by a cosine function. How do you combine these two motions into one smooth animation?

You can't just add them together. You need a way to translate one type of trigonometric motion into the other's language. This is where equivalent representations come in. They are like a Rosetta Stone for trigonometry, letting you rewrite sin(x) in terms of cos(x), or break down a complex angle like cos(75°) into simpler pieces you already know. In this lesson, we'll master these translations to simplify expressions and solve equations that look impossible at first glance.

A sine wave (blue) and a cosine wave (orange) illustrate their phase relationship.

Concept overview

flowchart TD
    A[Start: Given Trig Expression or Equation] --> B{What's the problem?};
    B --> C{Mixed functions like sin and cos²?};
    C --> D[Use Pythagorean Identity: sin²x + cos²x = 1];
    B --> E{Angle is a sum/difference like A+B or 75°?};
    E --> F[Use Sum/Difference Identity: sin(A+B) or cos(A-B)];
    B --> G{Angle is a multiple like 2x?};
    G --> H[Use Double-Angle Identity: sin(2x) or cos(2x)];
    D --> I[Simplify Algebraically];
    F --> I;
    H --> I;
    I --> J{Is it an equation to solve?};
    J -- Yes --> K[Solve for the variable x];
    J -- No --> L[End: Simplified Expression];
    K --> M[End: Solution Set for x];

Core explanation

Hey everyone, it's Saavi. Let's talk about one of the most powerful ideas in all of trigonometry: identities. An identity is just an equation that is true for all possible values of the variable. It’s a statement of fact, like saying 1 dollar is always equal to 4 quarters. We use these facts to swap out parts of an equation, making it simpler to work with.

The Cornerstone: The Pythagorean Identity

Everything starts with the unit circle. Remember that for any angle θ, the point on the unit circle is (cos θ, sin θ).


[[visual: The Pythagorean Identity on the Unit Circle]]

Imagine a right triangle inside that circle. The horizontal side has length cos θ, the vertical side has length sin θ, and because it's a unit circle, the hypotenuse (the radius) is always 1.

The Pythagorean Identity visualized on the unit circle.

Now, let's apply the good old Pythagorean Theorem: a² + b² = c².

For our triangle, that's (cos θ)² + (sin θ)² = 1².

Cleaning this up gives us the single most important identity in trigonometry:

sin²θ + cos²θ = 1

A quick note on notation: sin²θ is the standard way to write (sin θ)². This is to avoid confusion with sin(θ²), which is a totally different thing.

This identity is your Swiss Army knife. It lets you swap sines for cosines and vice-versa. If you know sin θ, you can find cos θ without ever knowing the angle θ itself.

Other Forms of the Pythagorean Identity

What happens if we take our main identity, sin²θ + cos²θ = 1, and divide every single term by cos²θ?

(sin²θ / cos²θ) + (cos²θ / cos²θ) = 1 / cos²θ

Remembering our quotient and reciprocal identities (tan θ = sin θ / cos θ and sec θ = 1 / cos θ), this simplifies to:

tan²θ + 1 = sec²θ

This is our second Pythagorean identity. It's incredibly useful when you're dealing with tangents and secants.

We can do the same thing by dividing the original identity by sin²θ to get the third form:

1 + cot²θ = csc²θ

You don't need to memorize the derivations every time, but you do need to know these three identities by heart. They are fundamental.

Combining and Separating Angles: Sum and Difference Identities

What if you need to find the sine or cosine of an angle that isn't one of our "nice" ones like 30°, 45°, or 60°? For example, how would you find the exact value of cos(75°) without a calculator?

You can't just add cos(30°) + cos(45°). That's not how functions work! This is where most students slip up. They try to distribute the cos or sin like it's a number. But sin(A + B) is NOT sin(A) + sin(B).

Instead, we use the sum and difference identities. These are the official formulas for finding the trig function of a combined angle.

Sum Identities:

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

Notice the pattern. The sine formula is a "mix" (sin-cos, cos-sin) and it keeps the same sign. The cosine formula is "pure" (cos-cos, sin-sin) but it flips the sign.

Difference Identities: How do we get the formula for sin(α - β)? We just think of it as sin(α + (-β)).

  • sin(α - β) = sin α cos(-β) + cos α sin(-β) Since cos(-β) = cos β (even function) and sin(-β) = -sin β (odd function), this becomes: sin(α - β) = sin α cos β - cos α sin β

  • Doing the same for cosine: cos(α - β) = cos α cos β + sin α sin β

From Sums to Doubles

A "double angle" like 2θ is just θ + θ. We can use the sum identities to find a formula for it.

Let's try it for sine: sin(2θ) = sin(θ + θ) = sin θ cos θ + cos θ sin θ sin(2θ) = 2 sin θ cos θ

This is the double-angle identity for sine. You can derive the one for cosine the same way, and it's a great way to see how these identities are all connected.

Why Bother? Solving Equations

Okay, that's a lot of formulas. But what's the point?

The point is to take messy equations and make them clean. When you're asked to solve a trigonometric equation, your goal is usually to get everything in terms of a single trig function. These identities are the tools you use to do that.

Imagine you have an equation like cos(2x) - sin(x) = 0. You have two different trig functions (cos and sin) and two different arguments (2x and x). It's a mess. But if you use an identity to replace cos(2x) with something that only involves sin(x), the equation suddenly becomes a simple quadratic that you already know how to solve.

Choosing the right identity is a skill, and it's a skill you build by practicing. Think of it like a puzzle. You have a jumble of pieces, and you're looking for the right identity to make them fit together.

Visualizing the identity tan²θ + 1 = sec²θ.

Worked examples

Let's put these identities into practice. The goal here isn't just to get the right answer, but to understand the strategy behind choosing the right tool for the job.

Example 1

Using the Pythagorean Identity to Solve an Equation

Problem: Find all solutions for 2cos²(x) + 3sin(x) - 3 = 0 on the interval [0, 2π).

Solution Walkthrough:

  1. 1
    Analyze the Problem
    Right away, you should notice a problem. The equation has both cos²(x) and sin(x). We can't solve it easily when it's mixed like this. Our goal is to rewrite the equation using only one trigonometric function.
  2. 2
    Choose the Right Tool
    We have a cos²(x). This should immediately make you think of the Pythagorean identity: sin²(x) + cos²(x) = 1. We can rearrange this to solve for cos²(x): cos²(x) = 1 - sin²(x)
  3. 3
    Substitute and Simplify
    Now, we replace cos²(x) in our original equation with 1 - sin²(x). 2(1 - sin²(x)) + 3sin(x) - 3 = 0

    Distribute the 2: 2 - 2sin²(x) + 3sin(x) - 3 = 0

    Combine the constant terms and reorder to make it look like a standard quadratic: -2sin²(x) + 3sin(x) - 1 = 0

    Let's multiply by -1 to make the leading term positive. It's just easier to look at. 2sin²(x) - 3sin(x) + 1 = 0

  4. 4
    Solve the Quadratic
    This is where students sometimes get nervous, but don't be! This is just a quadratic equation. To make it clearer, you can temporarily substitute a variable, say u = sin(x). 2u² - 3u + 1 = 0

    This factors nicely: (2u - 1)(u - 1) = 0

    So, u = 1/2 or u = 1.

  5. 5
    Substitute Back and Find x
    Now, we replace u with sin(x) again.
    • Case 1: sin(x) = 1. On the interval [0, 2π), this happens at x = π/2.
    • Case 2: sin(x) = 1/2. On the interval [0, 2π), this happens in Quadrant I and Quadrant II. The solutions are x = π/6 and x = 5π/6.

Final Answer: The solutions are x = π/6, x = π/2, and x = 5π/6.

Example 2

Using a Difference Identity to Find an Exact Value

Problem: Find the exact value of cos(15°).

Solution Walkthrough:

  1. 1
    Analyze the Problem
    15° is not one of our standard unit circle angles. We can't just look it up. The key word here is "exact value," which means no decimals from a calculator. This is a huge clue that we need to use an identity.
  2. 2
    Choose the Right Tool
    We need to express 15° as a sum or difference of angles we do know. The most obvious choice is 45° - 30°. Both 45° and 30° are standard angles. This means we need the cosine difference identity: cos(α - β) = cos α cos β + sin α sin β
  3. 3
    Substitute and Evaluate
    Let α = 45° and β = 30°. cos(15°) = cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°)

    Now, we just plug in the known values from the unit circle:

    • cos(45°) = √2 / 2
    • cos(30°) = √3 / 2
    • sin(45°) = √2 / 2
    • sin(30°) = 1 / 2

    cos(15°) = (√2 / 2)(√3 / 2) + (√2 / 2)(1 / 2)

  4. 4
    Simplify
    cos(15°) = (√6 / 4) + (√2 / 4) cos(15°) = (√6 + √2) / 4

Final Answer: The exact value is (√6 + √2) / 4.

Solutions for 2cos²(x) + 3sin(x) - 3 = 0 on [0, 2π).

Try it yourself

Ready to try a couple on your own? Remember to identify the type of problem first, then choose your identity.

  1. Solve for x on the interval [0, 2π): sin(x) - cos(2x) = 0

    Hint: You have two different functions (sin, cos) and two different arguments (x, 2x). Your first step must be to use an identity to fix this. The double-angle identity for cos(2x) has three versions. Which one will leave you with an equation that only contains sin(x)?*

  2. Find the exact value of sin(105°) without a calculator.

    Hint: How can you express 105° using two of the "famous" angles from the unit circle (like 30°, 45°, 60°, 90°)? Is it a sum or a difference? Once you figure that out, apply the correct identity.*

Good luck! Take your time and show your steps.

The function sin(x) - cos(2x) and its roots on [0, 2π).
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