What is the first step when a trigonometric equation contains both $\sin^2 x$ and $\cos x$?

Use the Pythagorean identity $\sin^2 x = 1 - \cos^2 x$ to rewrite the equation entirely in terms of $\cos x$. This creates a consistent quadratic form that can be solved using algebraic methods.

How does the number of potential solutions for a quadratic trig equation compare to a linear one in the range $0 \le x < 360^{\circ}$?

A linear trig equation typically has 2 solutions, whereas a quadratic trig equation can have up to 4. This is because the quadratic can produce two different values for the trig ratio, each potentially yielding two angles.

Why might one branch of a quadratic trigonometric solution be 'rejected'?

If the quadratic solution results in $\sin \theta = k$ or $\cos \theta = k$ where $|k| > 1$, it is rejected because the sine and cosine functions only output values between $-1$ and $1$. There are no real angles that satisfy such a ratio.

What is the danger of dividing an equation like $\sin^2 x = \sin x$ by $\sin x$?

Dividing by $\sin x$ assumes $\sin x \neq 0$, which causes you to lose the solutions where $\sin x = 0$ (e.g., $0^{\circ}, 180^{\circ}$). Instead, you should rearrange to $\sin^2 x - \sin x = 0$ and factorize.

When solving $\tan^2 x + 3\tan x + 2 = 0$, what is the benefit of using substitution?

Letting $u = \tan x$ transforms the expression into $u^2 + 3u + 2 = 0$. This allows you to use familiar algebraic tools like the quadratic formula or factorization without being distracted by the trigonometric notation.

Define a 'Quadratic Trigonometric Equation'.

It is an equation where a trigonometric function is the variable in a second-degree polynomial, typically appearing as $af(x)^2 + bf(x) + c = 0$. It requires solving for the trigonometric ratio before solving for the angle.

How do you find the second solution for $\cos x = k$ after finding the principal value $\alpha$?

For the cosine function, the second solution in the range $0$ to $360^{\circ}$ is found using $360^{\circ} - \alpha$. This is due to the symmetry of the cosine graph across the vertical lines at its peaks and troughs.

What identity is used to solve an equation containing $\tan^2 x$ and constants?

While $\tan^2 x$ can often be solved directly by square rooting, if it is mixed with other terms, you might use $\tan x = \frac{\sin x}{\cos x}$ or simply treat $\tan x$ as the quadratic variable $u$.

True or False: A quadratic equation in $\tan x$ can have solutions for any real value of $k$.

True. Unlike sine and cosine, the range of the tangent function is all real numbers $(-\infty, \infty)$, so any value of $k$ obtained from the quadratic will yield valid angles.

What is the 'Principal Value' in the context of solving these equations?

The principal value is the angle returned by a calculator's inverse trigonometric function. It serves as the starting point for finding all other solutions within the required domain using symmetry or periodicity.

Library Podcasts

Courses

Referral & Rewards

Revision Notes

AS-Level

Cambridge International Examinations

Maths

Pure 1

Trigonometry

Quadratic Simultaneous Equations

Quadratic Trigonometric Equations

Q: How does the number of potential solutions for a quadratic trig equation compare to a linear one in the range $0 \le x < 360^{\circ}$?

A linear trig equation typically has 2 solutions, whereas a quadratic trig equation can have up to 4. This is because the quadratic can produce two different values for the trig ratio, each potentially yielding two angles.

Q: Why might one branch of a quadratic trigonometric solution be 'rejected'?

If the quadratic solution results in $\sin \theta = k$ or $\cos \theta = k$ where $|k| > 1$, it is rejected because the sine and cosine functions only output values between $-1$ and $1$. There are no real angles that satisfy such a ratio.

Summary

Quadratic trigonometric equations are expressions where a trigonometric function is squared or appears in a quadratic form, such as $a\sin^2(x) + b\sin(x) + c = 0$ . Solving these requires a combination of algebraic factorization techniques and trigonometric identities to reduce the equation to simpler linear components, which are then solved within a specific numerical range.

1. Definition & Core Concepts

2. Underlying Principles

A flowchart showing the four main steps to solve quadratic trigonometric equations: using identities, substitution, solving the quadratic, and finding final angles.

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions

Quadratic Trigonometric Equations

Summary

1. Definition & Core Concepts

A quadratic trigonometric equation is an equation that can be written in the form $af(x)^2 + bf(x) + c = 0$ , where $f(x)$ represents a trigonometric function like $\sin(x)$ , $\cos(x)$ , or $\tan(x)$ .

Unlike linear trigonometric equations, these equations often involve squared terms (e.g., $\cos^2 \theta$ ) and can yield multiple sets of solutions because the quadratic part itself can produce two distinct trigonometric ratios.

The primary goal is to treat the trigonometric function as a single algebraic variable (substitution) to find the possible values of that ratio before finding the actual angles.

2. Underlying Principles

The Substitution Method is the logical foundation for solving these; by letting a variable like $u = \cos(x)$ , the equation transforms into a standard algebraic quadratic $au^2 + bu + c = 0$ .

The Pythagorean Identity, $\sin^2 \theta + \cos^2 \theta \equiv 1$ , is frequently used to ensure the entire equation is expressed in terms of a single trigonometric function (e.g., converting all terms to $\sin \theta$ ).

The principle of Null Factor Law applies here: once the quadratic is factorized into $(f(x) - k_1)(f(x) - k_2) = 0$ , we solve the two resulting linear equations $f(x) = k_1$ and $f(x) = k_2$ independently.