Trigonometry Test #03 Review, Spring 2010



[pic]

Section 5.1: Fundamental Identities

Reciprocal Identities:

[pic] [pic] [pic]

Quotient Identities:

[pic] [pic]

Pythagorean Identities:

[pic] [pic] [pic]

Negative-Angle Identities:

[pic] [pic] [pic]

[pic] [pic] [pic]

Section 5.2: Verifying Trigonometric Identities

Hints for Verifying Identities

• Learn (i.e., memorize) the fundamental identities from section 5.1, and be aware of their equivalent forms (like a re-arranged Pythagorean identity.

Reciprocal Identities:

[pic] [pic] [pic]

Quotient Identities:

[pic] [pic]

Pythagorean Identities:

[pic] [pic] [pic]

Negative-Angle Identities:

[pic] [pic] [pic]

[pic] [pic] [pic]

• Try to simplify the more complicated side until it looks like the simpler side.

• Sometimes it is helpful to express all trig functions in terms of sine and cosine, and then simplify.

• Usually it helps to factor when possible, and perform any indicated algebraic operations.

Example: replace [pic] with its factored form of [pic]

Example: replace [pic] with the fraction of [pic]

• As you make substitutions to convert one side into the other, always work toward the goal of the other side.

• A common trick is to multiply expressions like [pic]by the following fraction, because then a Pythagorean identity can be used to simplify:

[pic]

Section 5.3: Sum and Difference Identities for Cosine

Cosine of a Sum or Difference

[pic]

[pic]

Cofunction Identities

[pic] [pic]

[pic] [pic]

[pic] [pic]

Section 5.4: Sum and Difference Identities for Sine and Tangent

|Sine of a Sum or Difference |

|[pic] |

|[pic] |

|Tangent of a Sum or Difference |

|[pic] |

|[pic] |

Section 5.5: Double-Angle Identities

Double Angle Identities

[pic]

[pic]

[pic]

[pic]

[pic]

Product-To-Sum Identities

[pic]

[pic]

[pic]

[pic]

Sum-To-Product Identities

[pic]

[pic]

[pic]

[pic]

Section 5.6: Half-Angle Identities

Half-Angle Identities

[pic]

[pic]

[pic]

[pic]

[pic]

Section 6.1: Inverse Circular Functions

Vertical Line test

Any vertical line will intersect the graph of a function in at most one point.

Horizontal Line Test

Any horizontal line will intersect the graph of a one-to-one function in at most one point.

Inverse Function

The inverse function of a one-to-one function f is defined as [pic]. In other words, switch the x and y values of points on the graph of a function to obtain the graph of an inverse function.

Summary of Inverse Functions

• For a one-to-one function, each x-value corresponds to only one y-value, and each y-value corresponds to only one x-value (i.e., the function passes the vertical line test and the horizontal line test).

• If a function f is one-to-one, then f has an inverse function, which we write as f-1.

• The domain of f is the range of f-1, and the range of f is the domain of f-1.

• The graphs of f and f-1 are reflections of each other across the line y = x.

• To find f-1(x) from an algebraic function f(x), follow these steps:

o Interchange x and y in the equation y = f(x).

o Solve for y.

o The resulting expression that y is equal to is f-1(x).

Inverse Sine Function

[pic]or[pic]means that [pic], for [pic].

• Note that the domain of the sine function has to be restricted to make it a one-to-one function in order to define an inverse.

[pic]

|Graph of the Inverse Sine Function |

|[pic]OR[pic] |

| | |

|Domain: |Range: |

|Table of Values: | |

| |[pic] |

|x | |

|[pic] | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|Notes on the graph of the inverse sine function: |

|The inverse sine function is increasing and continuous on its domain. |

|Both the x- and y- intercepts are 0. |

|The inverse sine function is an odd function. |

Inverse Cosine Function

[pic]or[pic]means that [pic], for [pic].

• Note that the domain of the cosine function has to be restricted to make it a one-to-one function in order to define an inverse.

[pic]

|Graph of the Inverse Cosine Function |

|[pic]OR[pic] |

| | |

|Domain: |Range: |

|Table of Values: | |

| |[pic] |

|x | |

|[pic] | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|Notes on the graph of the inverse cosine function: |

|The inverse cosine function is decreasing and continuous on its domain. |

|Its x-intercept is 1, and its y-intercept is (/2. |

|The inverse cosine function is neither odd nor even. |

Inverse Tangent Function

[pic]or[pic]means that [pic], for [pic].

• Note that the domain of the tangent function has to be restricted to make it a one-to-one function in order to define an inverse.

[pic]

|Graph of the Inverse Tangent Function |

|[pic]OR [pic] |

| | |

|Domain: |Range: |

|Table of Values: | |

| |[pic] |

|x | |

|[pic] | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

| | |

|Notes on the graph of the inverse tangent function: |

|The inverse tangent function is increasing and continuous on its domain. |

|Both the x- and y- intercepts are 0. |

|The inverse tangent function is odd. |

|The lines [pic]are horizontal asymptotes. |

Inverse Cotangent, Secant, and Cosecant Functions

[pic]

[pic]

[pic]

Finding Inverse Trigonometric Functions with a Calculator

[pic] [pic] [pic]

Finding Trigonometric Functions of Inverse Trigonometric Functions

(Note: there are restrictions to the domains and ranges of the formulas below that are being glossed over…)

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

[pic] [pic] [pic]

Section 6.2: Trigonometric Equations I

To solve a trigonometric equation:

• If possible, graph the equation first so you can see what kind of answers to expect.

• Use identities so that only one of the trig functions occurs in the equation, and every occurrence of that trig function has the same argument.

o Sometimes factoring first can lead to “mini equations,” each of which has only one trig function, which means you don’t need to use trig identities.

• Use algebra to isolate the trig function

• Use inverse trig functions and the unit circle to find the angle or angles that solve the equation.

• Check your answers; sometimes an extraneous solution is introduced when you square both sides of the equation.

Section 6.3: Trigonometric Equations II

To solve a trigonometric equation:

• If possible, graph the equation first so you can see what kind of answers to expect.

• Use identities so that only one of the trig functions occurs in the equation, and every occurrence of that trig function has the same argument.

o Sometimes factoring first can lead to “mini equations,” each of which has only one trig function, which means you don’t need to use trig identities.

• Use algebra to isolate the trig function

• Use inverse trig functions and the unit circle to find the angle or angles that solve the equation.

• Check your answers; sometimes an extraneous solution is introduced when you square both sides of the equation.

Section 6.4: Equations Involving Inverse Trigonometric Functions

To solve an inverse trigonometric equation:

• If possible, graph the equation first so you can see what kind of answers to expect.

• Use algebra to isolate one of the inverse trig functions.

• Take the corresponding trig function of both sides of the equation.

o If necessary, use angle sum or difference formulas.

o If necessary, use the trick from section 6.1 to compute an exact algebraic expression for the composition of an inverse trig function and a trig function.

• Check your answers; sometimes an extraneous solution is introduced when you square both sides of the equation.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download