Trigonometric Identities & Equations

1. Core Concept Overview & Intuition

Trigonometry begins with the measurement of angles and the ratios of the sides of a right-angled triangle. The course treats radian measure, trigonometric ratios and trigonometric identities as the foundation of the topic, then extends these ideas to double-angle and factor formulae, and finally to the inverse relations, the $t$-formula and the plotting of trigonometric graphs. The central purpose of the topic is to develop the ability to evaluate trigonometric ratios and their identities and to solve trigonometric equations.

The intuition behind an identity is that it is an equation in the angle which holds for every value of the angle for which both sides are defined. An identity is therefore a statement about the structure of the trigonometric functions rather than a condition to be solved. For example, the Pythagorean relation between sine and cosine expresses the fact that the point tracing the unit circle always has coordinates whose squares sum to one. Once this structural fact is established, it can be manipulated algebraically to produce further identities, to simplify expressions, and to transform sums of trigonometric terms into products and products into sums.

The factor formulae are best understood as a change of representation. A sum or difference of sines or cosines can be rewritten as a product of sines and cosines, and conversely a product can be rewritten as a sum or difference. This is useful because products are often easier to set equal to zero and solve, while sums are often easier to integrate or to compare term by term. The double-angle formulae are a special case of the compound-angle formulae in which the two angles are equal, and they in turn generate the half-angle relations used in the $t$-substitution.

The $t$-substitution, often called the Weierstrass substitution, is a device for converting a trigonometric equation or expression in a single angle into an algebraic equation in a new variable $t$. The value of this device is that it reduces a problem about periodic functions to a problem about rational functions, which can then be handled by ordinary algebraic methods. The final stage of the topic, solving trigonometric equations and describing general solutions, depends on recognising that trigonometric functions are periodic: a single solution within one period generates infinitely many solutions obtained by adding whole multiples of the period.

Throughout the topic, the earlier work on the straight line, the circle, polar coordinates, the parabola, the ellipse and the hyperbola provides the geometric setting. The circle in particular underlies the definition of the trigonometric ratios through the coordinates of a moving point, and the conic sections provide contexts in which trigonometric parameters appear.

2. Mathematical Formalization & Core Definitions

2.1 Radian Measure and Circular Functions

An angle may be measured in degrees or in radians. The radian measure of an angle is defined by the arc it subtends on a circle centred at the vertex. If an arc of length $s$ is subtended at the centre of a circle of radius $r$, then the angle $\theta$ in radians is given by

$$
\theta=\frac{s}{r}.
$$

Because the ratio of arc length to radius is dimensionless, the radian is a pure number. A complete revolution corresponds to an arc equal to the whole circumference $2\pi r$, so a complete revolution is $2\pi$ radians. Consequently $180^{\circ}=\pi$ radians, and the conversion between the two measures is governed by this relation.

The circular functions are defined by placing the angle at the centre of a circle and reading off the coordinates of the point where the terminal side of the angle meets the circle. For a point on the unit circle, the cosine of the angle is the horizontal coordinate and the sine of the angle is the vertical coordinate. The tangent is defined as the ratio of the sine to the cosine, and the reciprocal functions secant, cosecant and cotangent are defined as the reciprocals of cosine, sine and tangent respectively. These definitions make the trigonometric functions periodic and give them their characteristic signs in the four quadrants.

2.2 Fundamental Pythagorean and Quotient Identities

The fundamental Pythagorean identity follows from the definition of sine and cosine on the unit circle. Since the point $(\cos\theta,\sin\theta)$ lies on the circle of radius one, its coordinates satisfy

$$
\cos^{2}\theta+\sin^{2}\theta=1.
$$

Dividing this identity by $\cos^{2}\theta$ produces the relation involving tangent and secant, and dividing by $\sin^{2}\theta$ produces the relation involving cotangent and cosecant. These derived forms are consequences of the fundamental identity and are used whenever an expression contains squares of the reciprocal functions.

The quotient identities express tangent and cotangent as ratios of sine and cosine. The tangent is the sine divided by the cosine, and the cotangent is the cosine divided by the sine. These quotient relations allow any expression involving tangent or cotangent to be rewritten entirely in terms of sine and cosine, which is often the first step in proving an identity.

The identities for negative angles are also fundamental. Sine, tangent, cotangent and cosecant are odd functions, so changing the sign of the angle changes the sign of the function. Cosine and secant are even functions, so changing the sign of the angle leaves the function unchanged. These parity relations are used constantly when simplifying expressions and when reducing an angle to a reference angle in the first quadrant.

2.3 Compound-Angle and Double-Angle Formulas

The compound-angle formulae express the sine, cosine and tangent of a sum or difference of two angles in terms of the sines and cosines of the individual angles. The sine of a sum is the sine of the first times the cosine of the second plus the cosine of the first times the sine of the second, and the sine of a difference is obtained by changing the sign of the second term. The cosine of a sum is the cosine of the first times the cosine of the second minus the sine of the first times the sine of the second, and the cosine of a difference is obtained by changing the sign of the second term. The tangent of a sum or difference is expressed as a quotient involving the tangents of the individual angles.

The double-angle formulae are obtained from the compound-angle formulae by setting the two angles equal. The sine of double an angle is twice the sine of the angle times the cosine of the angle. The cosine of double an angle has three equivalent forms: the difference of the squares of cosine and sine, twice the square of the cosine minus one, and one minus twice the square of the sine. Only one of these forms needs to be remembered, because the other two are derived from it using the Pythagorean identity. The tangent of double an angle is expressed in terms of the tangent of the single angle.

2.4 Factor Formulas

The factor formulae, also called the sum-to-product and product-to-sum formulae, allow a sum or difference of sines or cosines to be converted into a product of sines and cosines, and a product of sines and cosines to be converted into a sum or difference. The sum of two sines is twice the sine of half the sum times the cosine of half the difference. The difference of two sines is twice the cosine of half the sum times the sine of half the difference. The sum of two cosines is twice the cosine of half the sum times the cosine of half the difference, and the difference of two cosines is minus twice the sine of half the sum times the sine of half the difference.

The product-to-sum forms are the reverse relations. The product of two sines is half the difference of the cosines of the sum and the difference. The product of two cosines is half the sum of the cosines of the sum and the difference. The product of a sine and a cosine is half the sum of the sines of the sum and the difference. These relations are used to reduce products to sums, which is often the key step in solving an equation or in proving an identity.

2.5 The Half-Angle t-Substitution

The $t$-substitution introduces the variable $t$ as the tangent of half the angle. With this choice, the sine, cosine and tangent of the angle can all be expressed as rational functions of $t$. The sine of the angle is twice $t$ divided by one plus $t$ squared. The cosine of the angle is one minus $t$ squared divided by one plus $t$ squared. The tangent of the angle is twice $t$ divided by one minus $t$ squared. These expressions are derived from the double-angle formulae applied to half the angle, together with the Pythagorean identity.

The value of the substitution is that any trigonometric equation in a single angle becomes an algebraic equation in $t$. Once the equation is solved for $t$, the corresponding values of the angle are recovered from the relation between $t$ and the angle. Because the tangent function is periodic, the recovery of the angle must account for the periodicity, and the general solution is obtained by adding whole multiples of the period.

2.6 Solving Trigonometric Equations and General Solutions

A trigonometric equation is an equation involving one or more trigonometric functions of an unknown angle. Unlike an identity, a trigonometric equation is satisfied only by particular values of the angle. The solution of a trigonometric equation proceeds by reducing the equation to a standard form, using identities to express everything in terms of a single function of a single angle where possible, and then solving the resulting algebraic equation for that function.

Because the trigonometric functions are periodic, a single solution within one period generates infinitely many solutions. The general solution is obtained by adding whole multiples of the period to a particular solution. For the sine and cosine functions the period is $2\pi$ radians, and for the tangent function the period is $\pi$ radians. The general solution must therefore be stated with an integer parameter, and the particular solutions within the required interval are obtained by choosing suitable values of that parameter.