9.1 Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric Expressions
3 min read•june 18, 2024
Trigonometric identities are crucial tools for simplifying complex expressions and solving equations. They reveal the relationships between different trig functions, allowing us to manipulate and transform them. Mastering these identities opens up new problem-solving strategies in trigonometry.
Verifying and simplifying trig expressions involves applying identities and algebraic techniques. We'll learn to use Pythagorean, reciprocal, and , as well as double-angle, half-angle, and sum/difference formulas. These skills are essential for tackling advanced trig problems.
Verifying Trigonometric Identities
Verification of fundamental trigonometric identities
1+tan2θ=sec2θ connects tangent and secant functions
1+cot2θ=csc2θ links cotangent and cosecant functions
Reciprocal identities define reciprocal relationships between trigonometric functions
sinθ=cscθ1 sine is the reciprocal of cosecant
cosθ=secθ1 cosine is the reciprocal of secant
tanθ=cotθ1 tangent is the reciprocal of cotangent
Quotient identities express trigonometric functions as ratios of other functions
tanθ=cosθsinθ tangent is the ratio of sine to cosine
cotθ=sinθcosθ cotangent is the ratio of cosine to sine
Steps to verify an identity involve algebraic manipulation to show equivalence
Work with one side of the equation to simplify the expression
Apply trigonometric identities to transform the expression
Manipulate the expression algebraically until it matches the other side of the equation
Simplifying Trigonometric Expressions
Simplification of complex trigonometric expressions
Double-angle identities express trigonometric functions of double angles (2θ) in terms of the original angle (θ)
sin(2θ)=2sinθcosθ sine of double angle
cos(2θ)=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ cosine of double angle
tan(2θ)=1−tan2θ2tanθ tangent of double angle
Half-angle identities express trigonometric functions of half angles (2θ) in terms of the original angle (θ)
sin2(2θ)=21−cosθ square of sine of half angle
cos2(2θ)=21+cosθ square of cosine of half angle
tan(2θ)=sinθ1−cosθ=1+cosθsinθ tangent of half angle
Sum and difference identities express trigonometric functions of sums or differences of angles (α±β) in terms of the individual angles (α and β)
sin(α±β)=sinαcosβ±cosαsinβ sine of sum or difference
cos(α±β)=cosαcosβ∓sinαsinβ cosine of sum or difference
tan(α±β)=1∓tanαtanβtanα±tanβ tangent of sum or difference
Relationships between trigonometric functions
Inverse trigonometric functions undo the operation of the corresponding trigonometric function
sin−1(sinθ)=θ inverse sine of sine equals the original angle
cos−1(cosθ)=θ inverse cosine of cosine equals the original angle
tan−1(tanθ)=θ inverse tangent of tangent equals the original angle
Solving trigonometric equations involves applying identities and inverse functions
Simplify the equation using trigonometric identities to isolate the trigonometric function
Isolate the trigonometric function on one side of the equation
Apply the appropriate inverse trigonometric function to both sides
Solve for the variable, considering the domain and range of the functions involved
Express the solution in radians if necessary
Trigonometric Function Properties
Periodicity is a key characteristic of trigonometric functions
The fundamental period is the smallest positive value for which a function repeats
Understanding periodicity helps in analyzing trigonometric expressions and their behavior
Key Terms to Review (9)
Cosecant function: The cosecant function is the reciprocal of the sine function. It is defined as $\csc(x) = \frac{1}{\sin(x)}$ for all values of $x$ where $\sin(x) \ne 0$.
Cotangent function: The cotangent function, denoted as $\cot(x)$, is the reciprocal of the tangent function. It can be defined as $\cot(x) = \frac{1}{\tan(x)}$ or $\cot(x) = \frac{\cos(x)}{\sin(x)}$.
Even function: An even function is a type of function where $f(x) = f(-x)$ for all $x$ in its domain. Graphically, this means the function is symmetric with respect to the y-axis.
Even-odd identities: Even-odd identities are trigonometric identities that describe the symmetry properties of trigonometric functions. An even function satisfies $f(-x) = f(x)$, while an odd function satisfies $f(-x) = -f(x)$.
Odd function: An odd function is a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in its domain. This symmetry about the origin means that rotating the graph 180 degrees around the origin will produce the same graph.
Pythagorean identities: Pythagorean identities are fundamental trigonometric identities derived from the Pythagorean theorem. They relate the squares of the sine and cosine functions to 1, providing a basis for simplifying and verifying trigonometric expressions.
Quotient identities: Quotient identities are trigonometric identities that express tangent and cotangent functions as the quotient of sine and cosine functions. Specifically, $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ and $\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{\cos(\theta)}{\sin(\theta)}$.
Sine: Sine is a trigonometric function that represents the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle. It is denoted as $\sin(\theta)$ where $\theta$ is an angle.
Tangent function: The tangent function, denoted as $\tan(\theta)$, is a trigonometric function defined as the ratio of the sine and cosine functions: $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$. It is periodic with a period of $\pi$ radians.