IB AA: Trigonometric Functions

Trigonometric functions form the bridge between geometric relationships and algebraic analysis, a core pillar of the IB Mathematics: Analysis and Approaches (AA) curriculum. Moving beyond simple right-triangle ratios, you will learn to define these functions using the dynamic unit circle, wield powerful identities to simplify complex expressions, and solve sophisticated equations that model periodic phenomena. Mastering this topic is essential for calculus, complex numbers, and any field involving waves, oscillations, or circular motion.

From Degrees to Radians and the Unit Circle

The journey begins by abandoning degrees for a more mathematically natural measure: the radian. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius. Since the circumference of a circle is $2\pi r$, a full rotation corresponds to $2\pi$ radians. This gives the conversion: $180^{\circ }=\pi \text{ radians}$. Radians simplify calculus and directly link angular measure to arc length, given by $s=r\theta$, where $\theta$ is in radians.

This leads to the foundational model: the unit circle, a circle with radius 1 centered at the origin. Any angle $\theta$ (in standard position) corresponds to a point $P$ on this circle. The unit circle definitions of the primary trigonometric functions are then:

• The cosine of $\theta$, written $\cos \theta$, is the x-coordinate of point $P$.
• The sine of $\theta$, written $\sin \theta$, is the y-coordinate of point $P$.
• The tangent of $\theta$, written $\tan \theta$, is the ratio $\frac{\sin \theta }{\cos \theta }$ (the slope of the terminal side).

This definition extends the domain of trig functions to all real numbers, allows for negative values, and clearly illustrates key properties like periodicity and symmetry.

The Power of Trigonometric Identities

Identities are equations true for all values in the domain. They are your primary tools for simplification and transformation. The most fundamental set arises directly from the unit circle and Pythagoras' theorem: the Pythagorean identities. $${\sin }^2\theta +{\cos }^2\theta =1$$ Dividing this by ${\cos }^2\theta$ or ${\sin }^2\theta$ yields two other essential forms: $$1+{\tan }^2\theta ={\sec }^2\theta \text{and}1+{\cot }^2\theta ={\csc }^2\theta$$

For combining angles, you use the compound angle formulas: $$\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$$ $$\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$$ $$\tan (A\pm B)=\frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$$

A direct and incredibly useful application of these is the double angle formulas, derived by setting $B=A$ in the compound formulas: $$\sin 2A=2\sin A\cos A$$ $$\cos 2A={\cos }^{2}A-{\sin }^{2}A=2{\cos }^{2}A-1=1-2{\sin }^{2}A$$ $$\tan 2A=\frac{2\tan A}{1-{\tan }^{2}A}$$ The alternate forms for $\cos 2A$ are crucial for solving integrals and simplifying expressions involving squared trig functions.

Solving Trigonometric Equations

Solving equations like $2\cos (3x)-1=0$ requires a systematic approach. First, isolate the trig function: $\cos (3x)=\frac{1}{2}$. Next, consider the argument. Let $\theta =3x$. Solve $\cos \theta =\frac{1}{2}$ for $\theta$ using the unit circle or your calculator's inverse function, remembering that cosine is positive in the first and fourth quadrants. This gives the principal solutions: $\theta =\frac{\pi }{3}$ and $\theta =2\pi -\frac{\pi }{3}=\frac{5\pi }{3}$.

Because trigonometric functions are periodic, you must add the period. For cosine, the period is $2\pi$, so the general solution is $\theta =\frac{\pi }{3}+2\pi n$ or $\theta =\frac{5\pi }{3}+2\pi n$, where $n\in \mathbb{Z}$. Finally, back-substitute: since $\theta =3x$, you get $3x=\frac{\pi }{3}+2\pi n$ or $3x=\frac{5\pi }{3}+2\pi n$. Solving for $x$ gives $x=\frac{\pi }{9}+\frac{2\pi n}{3}$ or $x=\frac{5\pi }{9}+\frac{2\pi n}{3}$. If the problem specifies a domain (e.g., $0\le x<2\pi$), you substitute integer values for $n$ until all solutions within that domain are found.

Graphical Analysis and Interpretation

The graphs of $y=\sin x$ and $y=\cos x$ are sinusoidal waves with a period of $2\pi$, an amplitude of 1, and a range of $[-1,1]$. The graph of $y=\tan x$ has a period of $\pi$ and vertical asymptotes where $\cos x=0$. You must be able to interpret and sketch transformations of these base graphs. For $y=a\sin (b(x-c))+d$:

• $|a|$ is the amplitude (vertical stretch).
• $b$ affects the period, calculated as $\frac{2\pi }{|b|}$ for sine/cosine.
• $c$ is the phase shift (horizontal translation).
• $d$ is the vertical translation, setting the equation of the central axis.

Graphical analysis allows you to connect an equation's parameters to its visual features and vice-versa, which is vital for modeling real-world situations.

Common Pitfalls

1. Confusing Degrees and Radians: The most frequent error is having your calculator in the wrong mode. IB exams primarily use radians. Always check the mode when evaluating inverse functions or solving equations. If an angle is given in degrees, convert it to radians for calculus operations.
2. Incorrect General Solutions: Forgetting to add the period to all solutions or applying the wrong period is a major trap. Remember: the period of $\sin$ and $\cos$ is $2\pi$, while for $\tan$ it is $\pi$. The general solution must account for the function's full periodicity.
3. Misapplying Algebraic Rules to Arguments: You cannot distribute a function over addition inside the argument. $\sin (2x+\pi )$ is not equal to $\sin (2x)+\sin (\pi )$. You must use a compound angle formula to expand such expressions correctly.
4. Overlooking Domain Restrictions with Tangent: When solving equations involving $\tan x$ or when using identities like $1+{\tan }^{2}x={\sec }^{2}x$, you must remember that these are undefined where $\cos x=0$. Any proposed solution that makes the original denominator zero must be excluded.

Summary

• Radian measure is the standard angular measure in higher mathematics, directly linking angle to arc length via $s=r\theta$.
• The unit circle provides the definitive definitions for $\sin \theta$, $\cos \theta$, and $\tan \theta$ for all real angles, extending their domains and clarifying their properties.
• Core identities—Pythagorean, compound angle, and double angle—are essential tools for simplifying expressions, solving equations, and performing calculus operations.
• Solving trigonometric equations requires isolating the trig function, finding all angular solutions using the unit circle and periodicity, and then solving for the original variable.
• Understanding the graphs of trigonometric functions and their transformations (amplitude, period, phase shift) is key to connecting algebraic and graphical representations, a fundamental skill in analysis and modeling.