7.1 Solving Trigonometric Equations with Identities

Trigonometric Identities and Equations

Trigonometric identities are equations that hold true for all values in their domain. They let you rewrite complex trig expressions into simpler forms, which is the foundation for solving trig equations. This section covers the core identities you need to know and how to apply them when solving equations and proving equivalences.

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Trigonometric Identities and Equations

Application of fundamental trigonometric identities

There are several families of identities, and each one captures a different relationship between trig functions. Knowing which family to reach for is half the battle when simplifying or solving.

Reciprocal identities flip a trig function into its reciprocal counterpart:

• $\csc \theta = \frac{1}{\sin \theta}$
• $\sec \theta = \frac{1}{\cos \theta}$
• $\cot \theta = \frac{1}{\tan \theta}$

These are most useful when you spot a $\csc$, $\sec$, or $\cot$ in an equation and want to convert everything to $\sin$ and $\cos$.

Quotient identities define tangent and cotangent as ratios:

• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\cot \theta = \frac{\cos \theta}{\sin \theta}$

Pythagorean identities relate the squares of trig functions. These show up constantly:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $1 + \tan^2 \theta = \sec^2 \theta$
• $1 + \cot^2 \theta = \csc^2 \theta$

The first one is the most fundamental. The other two are derived from it by dividing through by $\cos^2 \theta$ or $\sin^2 \theta$, respectively. You can also rearrange any of these (for example, $\sin^2 \theta = 1 - \cos^2 \theta$) to make substitutions.

Even-odd identities describe how trig functions behave with negative inputs:

• $\sin(-\theta) = -\sin \theta$ (odd function, symmetric about the origin)
• $\cos(-\theta) = \cos \theta$ (even function, symmetric about the y-axis)
• $\tan(-\theta) = -\tan \theta$ (odd function)

When simplifying or solving, the general strategy is to apply identities to reduce the number of different trig functions in the expression, combine like terms, and cancel common factors until you reach a simpler form.

Manipulation of trigonometric identities

Proving that two trig expressions are equivalent (verifying an identity) follows a different approach than solving an equation. You don't solve for a variable; instead, you transform one or both sides until they match.

1. Work each side independently. Pick the more complicated side to simplify first. Do not move terms across the equals sign.

2. Convert to sine and cosine. This gives you a common language and makes patterns easier to spot.

3. Apply identities strategically. Look for Pythagorean substitutions, opportunities to factor, or common denominators to combine fractions.

4. Use algebra. Factor expressions (e.g., $\sin^2 \theta - 1 = (\sin \theta - 1)(\sin \theta + 1)$), multiply by conjugates, or combine fractions over a common denominator.

5. Show the two sides are equal. Once the simplified LHS matches the simplified RHS, the identity is verified.

A common mistake is treating a verification like an equation and performing operations on both sides simultaneously. Keep the sides separate.

Relationships of supplementary angle functions

These identities relate trig functions of an angle $\theta$ to functions of its supplement $\pi - \theta$. They come directly from the unit circle: the point at angle $\pi - \theta$ has the same y-coordinate but the opposite x-coordinate as the point at angle $\theta$.

• $\sin(\pi - \theta) = \sin \theta$
• $\cos(\pi - \theta) = -\cos \theta$
• $\tan(\pi - \theta) = -\tan \theta$

These are especially helpful when solving equations that produce reference angles in different quadrants. For instance, if $\sin \theta = 0.5$, the solutions in $[0, 2\pi)$ are $\theta = \frac{\pi}{6}$ and $\theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$, precisely because sine has the same value at supplementary angles.

Advanced Trigonometric Identities

These identities extend the toolkit to handle more complex expressions:

• Sum and difference identities express functions like $\sin(\alpha + \beta)$ or $\cos(\alpha - \beta)$ in terms of $\sin$ and $\cos$ of the individual angles.
• Double angle formulas relate functions of $2\theta$ to functions of $\theta$ (e.g., $\sin(2\theta) = 2\sin\theta\cos\theta$).
• Half angle formulas express functions of $\frac{\theta}{2}$ in terms of functions of $\theta$.

You'll typically use these when an equation mixes different angle multiples (like $\theta$ and $2\theta$ in the same equation) and you need everything in terms of a single angle.

Solving Trigonometric Equations

Apply fundamental trigonometric identities to solve equations

Solving a trig equation means finding all angles that make the equation true. Here's a reliable process:

1. Identify which identity applies. Look at the functions in the equation. If you see $\sec^2 \theta$, think Pythagorean. If you see $\tan \theta$ mixed with $\sin$ and $\cos$, think quotient identity.
2. Use the identity to rewrite the equation so it involves fewer trig functions, ideally just one.
3. Solve algebraically for that trig function (e.g., get $\sin \theta = \frac{1}{2}$).
4. Find all solutions in the given interval. Use the unit circle to identify every angle where the function takes that value. For example, $\sin \theta = \frac{1}{2}$ gives $\theta = \frac{\pi}{6}$ and $\theta = \frac{5\pi}{6}$ on $[0, 2\pi)$.
5. Check for extraneous solutions. If you squared both sides or multiplied by an expression that could be zero, plug your answers back in to verify.

Watch out for quadrantal angles (multiples of $\frac{\pi}{2}$). At these angles, some trig functions are undefined (e.g., $\tan \frac{\pi}{2}$), so they can't be valid solutions even if the algebra produces them.

Use algebraic techniques to solve trigonometric equations

Many trig equations look like algebra problems in disguise once you apply the right identity. The key algebraic techniques are:

• Factoring. If you can write the equation as a product equal to zero, set each factor to zero separately. For example, $2\sin^2\theta - \sin\theta = 0$ factors as $\sin\theta(2\sin\theta - 1) = 0$, giving $\sin\theta = 0$ or $\sin\theta = \frac{1}{2}$.
• Substitution. For equations like $2\cos^2\theta + 3\cos\theta - 2 = 0$, let $u = \cos\theta$ and solve the quadratic $2u^2 + 3u - 2 = 0$. Then convert back.
• Common denominators. When fractions with trig functions appear, combine them over a common denominator before solving.
• Inverse trig functions. Once you isolate the trig function (e.g., $\cos\theta = -\frac{\sqrt{3}}{2}$), use the unit circle or inverse functions to find the reference angle, then determine all solutions in the correct quadrants.

For the general solution, account for periodicity. Sine and cosine repeat every $2\pi$, so you add $2\pi k$ (where $k$ is any integer) to each solution. Tangent repeats every $\pi$, so you add $\pi k$. If the problem specifies an interval like $[0, 2\pi)$, list only the solutions that fall within it.

The unit circle is your best friend throughout this process. Visualizing where each trig function is positive, negative, or zero helps you catch all solutions and avoid missing any.