Trigonometric Identities and Equations: Study Guide

Trigonometric Identities and Equations

Core Trigonometric Formulas

Understanding and memorizing fundamental trigonometric identities is essential for solving equations and verifying identities in precalculus. These formulas allow for the transformation and simplification of trigonometric expressions.

• Sum and Difference Formulas:

  • Sum (sine): $\sin(A+B) = \sin A \cos B + \cos A \sin B$

  • Difference (sine): $\sin(A-B) = \sin A \cos B - \cos A \sin B$

  • Sum (cosine): $\cos(A+B) = \cos A \cos B - \sin A \sin B$

  • Difference (cosine): $\cos(A-B) = \cos A \cos B + \sin A \sin B$

  • Sum (tangent): $\tan(A+B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$

  • Difference (tangent): $\tan(A-B) = \dfrac{\tan A - \tan B}{1 + \tan A \tan B}$

• Double-Angle Formulas:

  • Sine: $\sin 2\theta = 2 \sin \theta \cos \theta$

  • Cosine: $\cos 2\theta = \cos^2 \theta - \sin^2 \theta = 2\cos^2 \theta - 1 = 1 - 2\sin^2 \theta$

  • Tangent: $\tan 2\theta = \dfrac{2\tan \theta}{1 - \tan^2 \theta}$

• Half-Angle Formulas:

  • Cosine: $\cos\left(\dfrac{\theta}{2}\right) = \pm \sqrt{\dfrac{1 + \cos \theta}{2}}$

  • Sine: $\sin\left(\dfrac{\theta}{2}\right) = \pm \sqrt{\dfrac{1 - \cos \theta}{2}}$

• Pythagorean Identities:

  • $\sin^2 \theta + \cos^2 \theta = 1$

  • $1 + \tan^2 \theta = \sec^2 \theta$

  • $1 + \cot^2 \theta = \csc^2 \theta$

• Quotient and Reciprocal Identities:

  • $\tan \theta = \dfrac{\sin \theta}{\cos \theta}$

  • $\cot \theta = \dfrac{\cos \theta}{\sin \theta}$

  • $\sec \theta = \dfrac{1}{\cos \theta}$

  • $\csc \theta = \dfrac{1}{\sin \theta}$

  • $\cot \theta = \dfrac{1}{\tan \theta}$

• Even-Odd Properties:

  • Even: $\cos(-x) = \cos x$, $\sec(-x) = \sec x$

  • Odd: $\sin(-x) = -\sin x$, $\tan(-x) = -\tan x$, $\csc(-x) = -\csc x$, $\cot(-x) = -\cot x$

Identity Verification Strategy

When verifying trigonometric identities, a systematic approach ensures accuracy and efficiency.

• Start with the more complicated side and aim to transform it to match the simpler side.

• Apply sum/difference formulas if you see expressions like $(x + \pi)$ or $(3\pi/2 - \theta)$.

• Rewrite secant or cosecant in terms of sine and cosine early if they appear in fractions.

• Combine fractions over a common denominator before simplifying.

• Expand numerators using FOIL or binomial expansion when appropriate.

• Look for Pythagorean patterns (e.g., $1 - \cos^2 \theta = \sin^2 \theta$).

• Factor and cancel common terms in numerators and denominators.

• Evaluate known values directly when possible (e.g., $\sin(3\pi/2) = -1$).

Example: Verifying $\sec(-x) \cdot \sin x = \tan x$

• Use even-odd identity: $\sec(-x) = \sec x$

• Rewrite in terms of sine and cosine: $\sec x = 1/\cos x$

• Multiply: $\dfrac{1}{\cos x} \cdot \sin x = \dfrac{\sin x}{\cos x} = \tan x$

Example: Combining Fractions

Given $\dfrac{1 - \sec \theta}{\tan \theta} - \dfrac{\tan \theta}{1 - \sec \theta}$:

1. Combine over a common denominator: $\dfrac{(1-\sec \theta)^2 - \tan^2 \theta}{\tan \theta (1-\sec \theta)}$

2. Expand numerator: $1 - 2\sec \theta + \sec^2 \theta - \tan^2 \theta$

3. Apply identity: $\sec^2 \theta - \tan^2 \theta = 1$

4. Numerator simplifies to $2 - 2\sec \theta = 2(1 - \sec \theta)$

5. Cancel: $\dfrac{2(1-\sec \theta)}{\tan \theta (1-\sec \theta)} = \dfrac{2}{\tan \theta} = 2\cot \theta$

Sum and Difference Formula Problems

To find $\sin(\alpha + \beta)$, $\cos(\alpha + \beta)$, or $\tan(\alpha + \beta)$ when given values for $\sin \alpha$, $\cos \alpha$, $\sin \beta$, or $\cos \beta$ (and their quadrants):

1. Draw reference triangles for $\alpha$ and $\beta$ using the given values and quadrants.

2. Use the Pythagorean theorem to find missing sides.

3. Assign correct signs to all ratios based on the quadrant (ASTC rule: All Students Take Calculus).

4. Plug values into the sum/difference formulas.

5. For tangent, either divide $\sin(\alpha+\beta)$ by $\cos(\alpha+\beta)$ or use the tangent sum formula directly.

6. Rationalize denominators and reduce answers as needed.

Common Pythagorean Triples: 3-4-5, 5-12-13, 7-24-25, 8-15-17, 9-40-41, 20-21-29

Double-Angle Problems

Double-angle identities are used to find the sine, cosine, or tangent of $2\theta$ given information about $\theta$.

1. Find all trig ratios for $\theta$ using a reference triangle and quadrant information.

2. Use $\sin 2\theta = 2 \sin \theta \cos \theta$.

3. For $\cos 2\theta$, choose the formula that matches the known values.

4. For $\tan 2\theta$, use $\tan 2\theta = \dfrac{2\tan \theta}{1 - \tan^2 \theta}$ or $\tan 2\theta = \dfrac{\sin 2\theta}{\cos 2\theta}$.

Half-Angle Problems

Half-angle identities allow calculation of trigonometric values for angles not directly on the unit circle.

1. Express the target angle as half of a known angle (e.g., $75^\circ = 150^\circ/2$).

2. Apply the half-angle formula, using the cosine (or sine) of the known angle.

3. Determine the sign based on the quadrant of the half-angle.

4. Simplify and rationalize as needed.

Solving Trigonometric Equations: General Solutions

Solving trigonometric equations involves finding all possible solutions, often using reference angles and periodicity.

• Identify the reference angle using the unit circle.

• List all solutions in the specified interval where the function has the given sign.

• Add the period to each solution for the general solution:

  • Sine and cosine: period $2\pi$ ($+2\pi n$)

  • Tangent and cotangent: period $\pi$ ($+\pi n$)

• For multiple-angle equations (e.g., $\cos 4x = -1/2$):

  • Let $u = 4x$, solve for $u$, then substitute back and solve for $x$.

  • List all solutions in the requested interval by incrementing $n$.

• Quadratic-form equations (e.g., $2\sin^2 x + 3\sin x = -1$):

  • Rewrite as a quadratic in $\sin x$ or $\cos x$.

  • Factor and solve for the trigonometric function, then solve for $x$.

• Simple factoring equations (e.g., $\sin^2 x - 1/4 = 0$):

  • Solve for the trigonometric function, remembering both positive and negative roots.

Common Mistakes to Avoid

• Not rationalizing denominators or reducing answers to lowest terms.

• Forgetting the $\pm$ when taking square roots.

• Assigning the wrong sign to a trigonometric ratio due to quadrant errors (always check ASTC).

• For tangent sum problems, using the formula directly is often faster and less error-prone than dividing $\sin(\alpha+\beta)$ by $\cos(\alpha+\beta)$.

• For multiple-angle equations, ensure all solutions within the interval are listed, not just the first per family.

Summary Table: Key Trigonometric Identities

Identity Type	Formula
Sum (sine)	$\sin(A+B) = \sin A \cos B + \cos A \sin B$
Difference (sine)	$\sin(A-B) = \sin A \cos B - \cos A \sin B$
Sum (cosine)	$\cos(A+B) = \cos A \cos B - \sin A \sin B$
Difference (cosine)	$\cos(A-B) = \cos A \cos B + \sin A \sin B$
Double Angle (sine)	$\sin 2\theta = 2 \sin \theta \cos \theta$
Double Angle (cosine)	$\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
Pythagorean	$\sin^2 \theta + \cos^2 \theta = 1$
Quotient	$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$
Reciprocal	$\sec \theta = \dfrac{1}{\cos \theta}$, $\csc \theta = \dfrac{1}{\sin \theta}$
Even/Odd	$\cos(-x) = \cos x$, $\sin(-x) = -\sin x$

Additional info: The ASTC rule refers to the signs of trigonometric functions in each quadrant: All (QI), Sine (QII), Tangent (QIII), Cosine (QIV) are positive. Rationalizing denominators and reducing answers is a standard requirement for final answers in trigonometric problems.