Glossary

7.1 Solving Trigonometric Equations with Identities

3 min read•june 25, 2024

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 when determining valid answers.

Trigonometric Identities and Equations

Application of fundamental trigonometric identities

Top images from around the web for Application of fundamental trigonometric identities

  • TrigCheatSheet.com: Identities View original

    Is this image relevant?

  • Example of Solving trigonometric equations involving trigonometric basic identities - Math2ever ... View original

    Is this image relevant?

  • TrigCheatSheet.com: Identities View original

    Is this image relevant?

1 of 3

Top images from around the web for Application of fundamental trigonometric identities

  • TrigCheatSheet.com: Identities View original

    Is this image relevant?

  • Example of Solving trigonometric equations involving trigonometric basic identities - Math2ever ... View original

    Is this image relevant?

  • TrigCheatSheet.com: Identities View original

    Is this image relevant?

1 of 3
  • express the reciprocal relationship between trigonometric functions
    • csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} defines as the reciprocal of
    • sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta} defines as the reciprocal of
    • cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta} defines as the reciprocal of
  • express one trigonometric function as the quotient of two others
    • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta} defines tangent as the ratio of sine to cosine
    • cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta} defines cotangent as the ratio of cosine to sine
  • relate the squares of trigonometric functions
    • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1 shows the sum of squared sine and cosine equals 1
    • 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta relates the squares of tangent and secant
    • 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta relates the squares of cotangent and cosecant
  • describe the symmetry of trigonometric functions
    • sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \theta indicates sine is an odd function (symmetric about origin)
    • cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \theta indicates cosine is an even function (symmetric about y-axis)
    • tan⁡(−θ)=−tan⁡θ\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 () and right-hand side () 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

  • relate trigonometric functions of angles that are supplements (π−θ\pi - \theta)
    • sin⁡(π−θ)=sin⁡θ\sin(\pi - \theta) = \sin \theta shows sine is the same for
    • cos⁡(π−θ)=−cos⁡θ\cos(\pi - \theta) = -\cos \theta shows cosine is the negative for supplementary angles
    • tan⁡(π−θ)=−tan⁡θ\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 (θ\theta)
  • Solve equations by applying opposite angle identities to transform the equation into a more manageable form

Advanced Trigonometric Identities

  • allow expressing trigonometric functions of sums or differences of angles
  • relate trigonometric functions of an angle to functions of twice that angle
  • 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 , which occur at multiples of 90° or π/2

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 to both sides of the equation to solve for the variable
  • Determine the by considering the periodicity of the trigonometric functions and the given interval
  • Use the to visualize solutions and understand their relationship to angles in radians

Key Terms to Review (25)

Cosecant: The cosecant (csc) is one of the six fundamental trigonometric functions, defined as the reciprocal of the sine function. It represents the ratio of the hypotenuse to the opposite side of a right triangle, and is used to describe the relationship between the sides and angles of a triangle.
Cosine: Cosine is a trigonometric function that represents the ratio of the adjacent side to the hypotenuse of a right triangle. It is one of the fundamental trigonometric functions, along with sine and tangent, and is essential in understanding and applying various concepts in mathematics, physics, and engineering.
Cotangent: The cotangent is one of the fundamental trigonometric functions, defined as the reciprocal of the tangent function. It represents the ratio of the adjacent side to the opposite side of a right triangle, providing a way to describe the relationship between the sides of a triangle and the angles within it.
Double Angle Formulas: Double angle formulas are trigonometric identities that express the sine, cosine, and tangent of twice an angle in terms of the original angle. These formulas are crucial for solving trigonometric equations and manipulating expressions involving sums and differences of angles.
Even-Odd Identities: Even-odd identities are a set of trigonometric identities that describe the behavior of even and odd trigonometric functions. These identities help simplify and solve trigonometric equations by exploiting the symmetry properties of these functions.
Fundamental Trigonometric Identities: Fundamental trigonometric identities are a set of basic equations that describe the relationships between the trigonometric functions, such as sine, cosine, and tangent. These identities are foundational to understanding and working with trigonometry, as they provide the framework for solving various trigonometric problems, including equations and expressions.
General Solution: The general solution of a trigonometric equation is the set of all possible solutions that satisfy the equation, including both the principal solution and the infinite number of other solutions that differ by multiples of the period of the function.
Half Angle Formulas: Half angle formulas are a set of trigonometric identities that allow for the simplification and evaluation of trigonometric expressions involving half-angles, or angles that are half the size of the original angle. These formulas are particularly useful in solving trigonometric equations and working with sum and difference identities.
Inverse Trigonometric Functions: Inverse trigonometric functions are the inverse operations of the standard trigonometric functions, allowing us to determine the angle given the ratio of the sides of a right triangle. They are essential in understanding and solving various trigonometric equations and problems.
LHS: LHS, or Left-Hand Side, is a term used in the context of solving trigonometric equations with identities. It refers to the expression on the left side of an equation, which is typically a trigonometric function or a combination of trigonometric functions.
Opposite Angle Functions: Opposite angle functions, also known as co-functions, are a set of trigonometric functions that have a specific relationship with each other. They are defined by the angles that are 90 degrees apart, or complementary angles, on the unit circle.
Opposite Angle Identities: Opposite angle identities are trigonometric identities that relate the trigonometric functions of an angle to the trigonometric functions of the angle's opposite. These identities are crucial in solving trigonometric equations and simplifying trigonometric expressions.
Periodicity: Periodicity refers to the recurring or cyclical nature of certain phenomena, where a pattern or sequence repeats itself at regular intervals. This concept is particularly relevant in the context of trigonometric functions, polar coordinates, and parametric equations, as it describes the periodic behavior of these mathematical constructs.
Pythagorean Identities: Pythagorean identities are fundamental trigonometric identities that describe the relationships between the trigonometric functions of a given angle. These identities are derived from the Pythagorean theorem and are essential in solving trigonometric equations and simplifying trigonometric expressions.
Quadrantal Angles: Quadrantal angles are special angles that are multiples of 90 degrees, or $\pi/2$ radians. These angles are significant in the study of trigonometry as they have unique properties and relationships with the trigonometric functions.
Quotient Identities: Quotient identities are a set of trigonometric identities that relate the trigonometric functions by expressing one function as a ratio or quotient of two other functions. These identities are particularly useful in simplifying trigonometric expressions and solving trigonometric equations.
Radians: Radians are a unit of measurement used to quantify angles, where one radian is the angle subtended by an arc on a circle that is equal in length to the radius of that circle. Radians are a fundamental concept in trigonometry and are essential for understanding and solving trigonometric equations.
Reciprocal Identities: Reciprocal identities are a set of trigonometric identities that relate the reciprocals of the trigonometric functions. These identities provide a way to express one trigonometric function in terms of another, which can be useful in solving trigonometric equations and simplifying trigonometric expressions.
RHS: RHS, or Right-Hand Side, is a term used in the context of solving trigonometric equations using identities. It refers to the expression on the right side of the equation, which is often compared to the Left-Hand Side (LHS) to determine the validity of the equation and find the solutions.
Secant: A secant is a line that intersects a curve or circle at two distinct points. It is a fundamental concept in trigonometry, geometry, and calculus, with applications in various fields of mathematics and physics.
Sine: The sine function is one of the fundamental trigonometric functions that describes the ratio of the length of the opposite side to the length of the hypotenuse of a right triangle. It is a periodic function that is widely used in various areas of mathematics, physics, and engineering.
Sum and Difference Identities: Sum and difference identities are trigonometric identities that describe the relationship between the trigonometric functions of the sum or difference of two angles. These identities are essential in solving trigonometric equations and simplifying trigonometric expressions.
Supplementary Angles: Supplementary angles are two angles that, when added together, equal 180 degrees. They are angles that form a straight line or a flat surface. This concept is important in understanding the relationships between angles and their properties, which is crucial in the context of topics such as 5.1 Angles, 7.1 Solving Trigonometric Equations with Identities, and 7.3 Double-Angle, Half-Angle, and Reduction Formulas.
Tangent: A tangent is a line that touches a curve at a single point, forming a right angle with the curve at that point. It is a fundamental concept in trigonometry, geometry, and calculus, with applications across various mathematical and scientific disciplines.
Unit Circle: The unit circle is a circle with a radius of 1 unit, centered at the origin (0,0) of the coordinate plane. It is a fundamental tool in trigonometry that helps visualize and understand the relationships between angles, their trigonometric functions, and the coordinates of points on the circle.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide