UK A-Level: Trigonometry Identities

Trigonometric identities are the essential toolkit for manipulating and solving equations in A-Level Mathematics and beyond. Mastering them unlocks problems in calculus, physics, and engineering, transforming complex expressions into solvable forms. This guide builds from foundational identities to advanced applications, equipping you with the systematic approach required for exam success.

The Foundational Pythagorean Set

All trigonometric manipulation begins with the core Pythagorean identity, derived directly from the unit circle. The fundamental relationship is: $${\sin }^2\theta +{\cos }^2\theta \equiv 1$$ The notation "$\equiv$" denotes an identity, meaning it is true for all values of $\theta$, unlike an equation which is only true for specific values. From this single statement, two other useful forms can be derived by dividing through by ${\cos }^2\theta$ or ${\sin }^2\theta$.

Dividing by ${\cos }^2\theta$ gives: $${\tan }^2\theta +1\equiv {\sec }^2\theta$$ where $\sec \theta =\frac{1}{\cos \theta }$.

Dividing by ${\sin }^2\theta$ gives: $$1+{\cot }^2\theta \equiv {\mathrm{cosec}}^2\theta$$ where $\mathrm{cosec}\theta =\frac{1}{\sin \theta }$ and $\cot \theta =\frac{1}{\tan \theta }$.

You use these identities to simplify expressions, prove more complex identities, and, most crucially, to solve equations. For instance, if you encounter an equation with both ${\sin }^{2}x$ and ${\cos }^{2}x$, you can use the first identity to express the entire equation in terms of just one trigonometric function.

Addition and Double Angle Formulas

The addition formulas allow you to find the sine, cosine, or tangent of a sum or difference of two angles. They are not intuitively obvious but must be memorised:

$$\sin (A\pm B)\equiv \sin A\cos B\pm \cos A\sin B$$ $$\cos (A\pm B)\equiv \cos A\cos B\mp \sin A\sin B$$ $$\tan (A\pm B)\equiv \frac{\tan A\pm \tan B}{1\mp \tan A\tan B}$$

A critical application is deriving the double angle formulas. By setting $B=A$ in the addition formulas, you get: $$\sin 2A\equiv 2\sin A\cos A$$ $$\cos 2A\equiv {\cos }^{2}A-{\sin }^{2}A$$ The $\cos 2A$ identity has two other forms, obtained using ${\sin }^{2}A+{\cos }^{2}A=1$: $$\cos 2A\equiv 2{\cos }^{2}A-1$$ $$\cos 2A\equiv 1-2{\sin }^{2}A$$ These alternative forms are incredibly powerful for integrating ${\sin }^{2}x$ or ${\cos }^{2}x$ and for solving equations. The double angle formula for tangent is: $$\tan 2A\equiv \frac{2\tan A}{1-{\tan }^{2}A}$$

The Harmonic Form: Rsin(x ± α) and Rcos(x ± α)

A major technique involves expressing a combination like $a\sin x+b\cos x$ as a single sine or cosine function. This is called the harmonic form. The process uses the addition formulas in reverse.

Any expression $a\sin x+b\cos x$ can be written as either $R\sin (x+\alpha )$ or $R\cos (x-\alpha )$, where $R>0$ and $0^{\circ }<\alpha <90^{\circ }$ (or $0<\alpha <\pi /2$ radians). To find $R$ and $\alpha$:

1. Expand your target form using the addition formula. For $R\sin (x+\alpha )$:

$$R\sin (x+\alpha )=R\sin x\cos \alpha +R\cos x\sin \alpha$$

1. Equate coefficients with $a\sin x+b\cos x$:

$$a=R\cos \alpha ,b=R\sin \alpha$$

1. Find $R$ using Pythagoras: $R=\sqrt{a^2+b^2}$.
2. Find $\alpha$ using $\tan \alpha =\frac{b}{a}$, ensuring you select the quadrant for $\alpha$ correctly by checking the signs of $a$ and $b$.

This form is indispensable for finding the maximum/minimum values of expressions and for solving equations of the type $a\sin x+b\cos x=c$ easily.

Solving Trigonometric Equations

Solving equations requires careful, structured work. Your primary tools are the identities you've learned and a precise understanding of the unit circle.

1. Simplify the Equation: Use identities (Pythagorean, double angle, harmonic form) to get the equation in terms of a single trigonometric function of a single angle (e.g., $\sin 2\theta$, $\cos (\theta +30^{\circ })$).
2. Solve for the Basic Angle: Isolate the function and find the principal value (the calculator answer) and then all solutions within the given range. Remember:

• For $\sin \theta =k$, solutions are $\theta =\alpha ,180^{\circ }-\alpha$ (plus periodicity).
• For $\cos \theta =k$, solutions are $\theta =\alpha ,360^{\circ }-\alpha$ (plus periodicity).
• For $\tan \theta =k$, solutions are $\theta =\alpha ,180^{\circ }+\alpha$ (plus periodicity).

1. Apply the Periodicity: Extend your solutions using the period of the function: $360^{\circ }$ for $\sin$ and $\cos$, $180^{\circ }$ for $\tan$. In radians, the periods are $2\pi$ and $\pi$ respectively.
2. Adjust for the Compound Angle: If you solved for a compound angle like $3x-45^{\circ }$, remember to solve for $x$ itself at the end.

You must be fluent in both degrees and radians. Always check the question for the specified unit and adjust your calculator accordingly. A common exam instruction is to solve for $0\le x<2\pi$ (radians) or $0^{\circ }\le \theta <360^{\circ }$.

Small Angle Approximations

For very small angles measured in radians, trigonometric functions can be approximated by simple polynomial expressions. These are derived from calculus (Maclaurin series) and are exceptionally useful in applied mathematics and physics for simplifying models.

For a small angle $\theta$ (in radians): $$\sin \theta \approx \theta$$ $$\cos \theta \approx 1-\frac{{\theta }^2}{2}$$ $$\tan \theta \approx \theta$$

These approximations become more accurate as $\theta$ approaches zero. A typical use is to simplify an expression like $\frac{\sin 3\theta }{2\theta }$ when $\theta$ is small: it becomes approximately $\frac{3\theta }{2\theta }=\frac{3}{2}$. You must never mix units here—these approximations are only valid when the angle is in radians.

Common Pitfalls

1. Algebraic Errors with Squares: When taking the square root of both sides after using a Pythagorean identity (e.g., ${\sin }^2\theta =\frac{1}{4}$), you must remember the $\pm$ sign. $\sin \theta =\pm \frac{1}{2}$ leads to twice as many solutions in a given cycle. Forgetting the $\pm$ is a frequent source of lost marks.

1. Incorrect Quadrant Selection: When finding an angle $\alpha$ for the harmonic form or when solving an equation using inverse functions, relying solely on your calculator can mislead you. The calculator gives the principal value. You must use the signs of the relevant trigonometric ratios (ASTC or "All Students Take Chemistry" cast diagram) to determine all valid angles in the specified range.

1. Mishandling Compound Angles: If you solve an equation to find that $2x+\frac{\pi }{4}=\frac{\pi }{6}$, your final answer is not $x=\frac{\pi }{6}$. You must solve for $x$: $x=(\frac{\pi }{6}-\frac{\pi }{4})/2$. Furthermore, you must apply the periodicity to the compound angle before solving for $x$. If the period of $\sin (2x)$ is $\pi$, you add $n\pi$ to $2x$, not to $x$.

1. Confusing Degrees and Radians: Using the wrong mode on your calculator will yield a numerically wrong answer. Even worse, applying degree-based periodicity ($+360^{\circ }$) to an equation given in radians is a catastrophic error. Always double-check the unit specified in the question and write it clearly next to your answers.

Summary

• The Pythagorean identities (${\sin }^2\theta +{\cos }^2\theta \equiv 1$ and its derivatives) are fundamental for simplifying expressions and changing the form of equations.
• Addition and double angle formulas allow you to manipulate combinations of angles, with the $\cos 2A$ identities being particularly useful for integration and solving quadratic trigonometric equations.
• The harmonic form $R\sin (x\pm \alpha )$ lets you rewrite $a\sin x+b\cos x$ as a single wave, simplifying the process of finding maxima, minima, and solutions.
• Solving equations requires a methodical approach: simplify using identities, find the basic angle, apply periodicity, and finally solve for the original variable.
• For very small angles in radians, the small angle approximations ($\sin \theta \approx \theta$, $\cos \theta \approx 1-{\theta }^2/2$, $\tan \theta \approx \theta$) provide powerful tools for simplification in applied contexts.