Trig Ratios & Identities — SL 3.5 Trigonometric ratios and identities (tan; basic identities)

Long before calculators existed, ancient civilizations needed reliable ways to measure heights, distances, and angles. The word trigonometry literally means "triangle measurement" — from the Greek words trigonon (triangle) and metron (measure). Over thousands of years, mathematicians across many cultures developed the ratios and identities you will learn in this lesson. These tools turn angle information into side-length information and vice versa, which is essential for navigation, engineering, astronomy, and physics.

The central question this lesson addresses is: how do the three primary trig ratios — sine, cosine, and tangent — relate to each other, and what algebraic identities tie them together? Understanding these connections will let you simplify expressions, verify equations, and solve problems far more efficiently than treating each ratio in isolation.

In IB Math AA SL 3.5, you build on your existing knowledge of sine and cosine to formally define the tangent ratio and to learn the basic trigonometric identities. These identities are equations that are true for every allowed value of the angle, not just for special cases. Let's nail down the foundational ideas.

In the diagram above, the angle θ sits at the origin, and the point on the unit circle has coordinates (cos θ, sin θ). The vertical leg of the triangle is sin θ, and the horizontal leg is cos θ. Because the hypotenuse is 1, the Pythagorean theorem gives us cos²θ + sin²θ = 1 directly. The tangent is the ratio of the vertical to horizontal legs: tan θ = sin θ / cos θ. Notice that when cos θ = 0 (at 90° and 270°), the tangent is undefined — you'd be dividing by zero.

There are three identities at the heart of SL 3.5. Each one can be derived from the definitions of sine, cosine, and tangent, combined with the Pythagorean theorem. You should memorize these — they appear throughout the IB course.

You can combine these identities. For example, dividing the entire Pythagorean identity by cos²θ gives sin²θ/cos²θ + 1 = 1/cos²θ, which simplifies to tan²θ + 1 = sec²θ. While the secant function isn't heavily tested at SL, this derivation shows how all trig identities are connected. At SL level, focus on being fluent with the three boxed results above.

When you move beyond acute angles, the sign of each trig ratio depends on which quadrant the terminal arm of the angle falls in. The CAST diagram is a mnemonic that tells you which ratios are positive in each quadrant. Starting from the fourth quadrant and reading counterclockwise, the letters stand for Cosine, All, Sine, Tangent. Understanding CAST is critical for applying the Pythagorean identity, because that identity tells you the magnitude of a ratio but not its sign — you need the quadrant information for that.

Why does the sign pattern work this way? On the unit circle, cos θ is the x-coordinate and sin θ is the y-coordinate. In Quadrant II, x is negative but y is positive, so cos θ < 0 and sin θ > 0. Since tan θ = sin θ / cos θ, a positive divided by a negative gives a negative tangent. Run through each quadrant with this logic and the CAST diagram will make perfect sense.

A classic IB-style question gives you one trig ratio and the quadrant, then asks you to find the remaining ratios. Let's work through a full example.

Identity-proving questions regularly appear on IB exams. The golden rule is: work on one side only and transform it until it matches the other side. Never "cross the equals sign" — that would assume the identity is already true, which is what you're trying to show. Here are the main strategies, along with common pitfalls.

The tangent identity and the Pythagorean identity are the foundation for more advanced work in IB Math AA. At the HL level and in further mathematics, you'll encounter compound angle formulas, double angle formulas, and reciprocal trig functions (sec, csc, cot) — all of which build directly on the identities in this lesson. Even in calculus, when you differentiate and integrate trigonometric functions, these identities appear constantly.

Mastering SL 3.5 thoroughly will make the transition to these advanced topics much smoother. You'll recognize patterns, apply identities automatically, and focus your energy on the new concepts rather than struggling with the basics. Think of the identities in this lesson as the vocabulary of trigonometry — once you speak the language fluently, more complex conversations become possible.