What is an Identity?
Not every equation is true sometimes — some are true always. Learn to distinguish a conditional equation from a trigonometric identity, and discover why identities are the foundation of everything in this chapter.
Introduction
In algebra you solved equations like — true only when . In trigonometry, certain equations hold for every permissible value of the variable. These universal truths are called identities, and they are the tools you'll use to simplify, verify, and solve throughout this entire unit.
Past Knowledge
You know all six trig ratios and can evaluate them at any angle on the unit circle.
Today's Goal
Distinguish identities from conditional equations and understand why identities matter.
Future Success
Every lesson in Chapter 5 builds on the concept of an identity — this is your starting line.
Key Concepts
Conditional Equation vs. Identity
| Type | Example | True for… |
|---|---|---|
| Conditional Equation | Only specific values of | |
| Identity | All values of |
Why Do Identities Work?
On the unit circle, every point is and lies on the circle . Substituting gives — a statement baked into the geometry itself. No matter the angle, the point is always on the circle, so the equation is always true.
Visualizing: Both Sides Are the Same Curve
If an equation is an identity, graphing the left side and the right side should produce identical curves. Below, the blue curve is and the red dashed curve is . They overlap perfectly.
The Graphical Test
To check whether an equation might be an identity, graph both sides. If they overlap completely, the equation is likely an identity. If they differ at even one point, it is not an identity. (A formal proof is still needed for certainty.)
Categories of Trigonometric Identities
Throughout Chapter 5 you will learn several families:
| Family | Example | Lesson |
|---|---|---|
| Reciprocal / Quotient | 5.2 | |
| Pythagorean | 5.3 – 5.4 | |
| Even / Odd | 5.5 | |
| Cofunction | 5.6 | |
| Sum / Difference | 5.10 – 5.12 | |
| Double / Half-Angle | 5.13 – 5.14 |
Worked Examples
Identifying an Identity
Question: Is an identity or a conditional equation?
Step 1: Recall the definition of tangent from right-triangle trigonometry: .
Step 2: On the unit circle, (opposite) and (adjacent), giving .
Step 3: This relationship holds for every angle where , which is all permissible values.
Final Answer: It is an identity — it is true for all values of in its domain.
Disproving a Proposed Identity
Question: Is an identity?
Step 1: Test a specific value. Let .
Step 2: Evaluate the left side:
Step 3: Since , the equation fails at .
Final Answer: Not an identity. One counterexample is enough to disprove it.
Using the Graphical Test
Question: Use a graph to decide whether appears to be an identity.
Step 1: Graph the left side, , and the right side, .
Step 2: Observe that both curves overlap completely for all -values.
Step 3: This strongly suggests that is an identity. (You will formally prove this using the double-angle formulas in Lesson 5.13.)
Final Answer: The graphs coincide — the equation appears to be an identity (and it is, as you'll prove later).
Common Pitfalls
Checking Only One Value
Plugging in and seeing that both sides equal the same number does not prove an identity. An equation can be true at one angle and false at another. You need algebraic proof or a counterexample — never just one test value.
Confusing “Solving” with “Verifying”
Solving an equation means finding specific values that make it true. Verifying an identity means proving it holds for allvalues. These are fundamentally different tasks — don't mix up the approaches.
Real-Life Applications
Signal Processing & Electronics
Electrical engineers routinely use trig identities to simplify alternating-current (AC) circuit equations. For example, when two voltage signals are combined, identities let engineers rewrite the sum of sines as a single sinusoidal expression — making it far easier to predict peak voltage, phase, and frequency without brute-force calculation. Every smartphone, radio, and WiFi router relies on these identity-based simplifications.
Practice Quiz
Loading...