Trigonometric identities are key relationships between trig functions. They help simplify complex expressions and solve equations. By understanding these connections, you can tackle tricky problems and uncover hidden patterns in trigonometry.

Solving trig equations requires a mix of algebraic skills and trig knowledge. You'll use identities to rewrite expressions, isolate variables, and find solutions. Remember to consider function domains and periodicity when determining valid answers.

Trigonometric Identities and Equations

Application of fundamental trigonometric identities

• Reciprocal identities express the reciprocal relationship between trigonometric functions
  • $\csc \theta = \frac{1}{\sin \theta}$ defines cosecant as the reciprocal of sine
  • $\sec \theta = \frac{1}{\cos \theta}$ defines secant as the reciprocal of cosine
  • $\cot \theta = \frac{1}{\tan \theta}$ defines cotangent as the reciprocal of tangent
• Quotient identities express one trigonometric function as the quotient of two others
  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$ defines tangent as the ratio of sine to cosine
  • $\cot \theta = \frac{\cos \theta}{\sin \theta}$ defines cotangent as the ratio of cosine to sine
• Pythagorean identities relate the squares of trigonometric functions
  • $\sin^2 \theta + \cos^2 \theta = 1$ shows the sum of squared sine and cosine equals 1
  • $1 + \tan^2 \theta = \sec^2 \theta$ relates the squares of tangent and secant
  • $1 + \cot^2 \theta = \csc^2 \theta$ relates the squares of cotangent and cosecant
• Even-odd identities describe the symmetry of trigonometric functions
  • $\sin(-\theta) = -\sin \theta$ indicates sine is an odd function (symmetric about origin)
  • $\cos(-\theta) = \cos \theta$ indicates cosine is an even function (symmetric about y-axis)
  • $\tan(-\theta) = -\tan \theta$ indicates tangent is an odd function (symmetric about origin)
• Simplify expressions by applying appropriate identities to reduce complexity (combine like terms, cancel factors)
• Solve trigonometric equations by applying identities to isolate the variable and find solutions within the domain

Manipulation of trigonometric identities

• Simplify the left-hand side (LHS) and right-hand side (RHS) of the identity separately to create equivalent expressions
• Apply trigonometric identities to manipulate the LHS and RHS, transforming them into a common form
• Utilize algebraic techniques such as factoring, expanding, and finding common denominators to simplify expressions
• Verify the identity by showing that the simplified LHS equals the simplified RHS, confirming their equivalence

Relationships of opposite angle functions

• Opposite angle identities relate trigonometric functions of angles that are supplements ($\pi - \theta$)
  • $\sin(\pi - \theta) = \sin \theta$ shows sine is the same for supplementary angles
  • $\cos(\pi - \theta) = -\cos \theta$ shows cosine is the negative for supplementary angles
  • $\tan(\pi - \theta) = -\tan \theta$ shows tangent is the negative for supplementary angles
• Simplify expressions using opposite angle identities to rewrite functions in terms of the original angle ($\theta$)
• Solve equations by applying opposite angle identities to transform the equation into a more manageable form

Advanced Trigonometric Identities

• Sum and difference identities allow expressing trigonometric functions of sums or differences of angles
• Double angle formulas relate trigonometric functions of an angle to functions of twice that angle
• Half angle formulas express trigonometric functions of half an angle in terms of functions of the full angle
• These identities are useful for simplifying complex trigonometric expressions and solving equations

Solving Trigonometric Equations

Apply fundamental trigonometric identities to solve equations

• Identify the appropriate identity to apply based on the given equation (reciprocal, quotient, Pythagorean, even-odd)
• Manipulate the equation using the selected identity to isolate the variable and simplify the expression
• Solve the resulting equation for the variable, considering the domain of the trigonometric functions involved
• Determine valid solutions within the domain of the functions and the given interval (if applicable)
• Consider special cases such as quadrantal angles, which occur at multiples of 90° or π/2 radians

Use algebraic techniques to solve trigonometric equations

• Simplify the equation by applying trigonometric identities to rewrite functions and reduce complexity
• Utilize algebraic techniques such as factoring, expanding, and finding common denominators to manipulate the equation
• Isolate the trigonometric function containing the variable on one side of the equation
• Apply inverse trigonometric functions to both sides of the equation to solve for the variable
• Determine the general solution by considering the periodicity of the trigonometric functions and the given interval
• Use the unit circle to visualize solutions and understand their relationship to angles in radians