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.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related searches
- chapter 9 test review quizlet
- chapter 4 test review answers
- unit 2 test review answers
- unit 2 test review biology
- unit 2 test review math
- unit test review math
- right triangle trigonometry test answers
- protein synthesis test review answers
- unit 4 test review math answers
- naming compounds test review practice
- icd 10 review test results
- trigonometry practice test with answers