Quarter 2 Class Notes
UNIT 6: Trigonometry
1/18 & 28- Reciprocal Trig Functions and Trig IDs
*The reciprocal of sine is cosecant.
*The reciprocal of cosine is secant.
*The reciprocal of tangent is cotangent.
*Pythagorean Trig ID = sin^2 + cos^2 = 1
*The other 2 trig ID's can be found by dividing by either sin^2 or cos^2.
*The reciprocal of sine is cosecant.
*The reciprocal of cosine is secant.
*The reciprocal of tangent is cotangent.
*Pythagorean Trig ID = sin^2 + cos^2 = 1
*The other 2 trig ID's can be found by dividing by either sin^2 or cos^2.
1/16 & 17- Converting between Degrees & Radians
*There are pi radians in 180 degrees.
*To convert from degrees to radians, set up a proportion (180 degrees over pi radians).
*To convert from radians to degrees, simply substitute pi with 180 degrees.
*Be careful to round accordingly!
*There are pi radians in 180 degrees.
*To convert from degrees to radians, set up a proportion (180 degrees over pi radians).
*To convert from radians to degrees, simply substitute pi with 180 degrees.
*Be careful to round accordingly!
1/4- 1/14- Co-Terminal, Reference Angles
*On a coordinate axis we plot (x, y). In trigonometry, we plot (cos theta, sin theta).
*Co-terminal angles are angles that have the same terminal side.
*You find co-terminal angles by either adding or subtracting 360 degrees.
*Reference Angles are between 0 and 90 degrees. They are always drawn to the x-axis.
*Don't forget "All Students Take Calculus."
*To find the trig value of an angle bigger than 90 degrees, use a reference angle and determine the sign. Then use the chart to determine exact values.
*On a coordinate axis we plot (x, y). In trigonometry, we plot (cos theta, sin theta).
*Co-terminal angles are angles that have the same terminal side.
*You find co-terminal angles by either adding or subtracting 360 degrees.
*Reference Angles are between 0 and 90 degrees. They are always drawn to the x-axis.
*Don't forget "All Students Take Calculus."
*To find the trig value of an angle bigger than 90 degrees, use a reference angle and determine the sign. Then use the chart to determine exact values.
1/3- Special Right Triangles
*You MUST memorize the chart for the sine, cosine, and tangent of 0, 30, 45, 60 and 90 degrees.
*You MUST memorize the chart for the sine, cosine, and tangent of 0, 30, 45, 60 and 90 degrees.
1/2- Basic Right Triangle Trigonometry
*Pythagorean's Theorem (can only be used for right triangles): a^2 + b^2 = c^2. Used to find missing sides.
*Trigonometry- the study between the relationship between sides and angles of a triangle.
*SOH CAH TOA: the ratios used to find either missing sides or angles in a right triangle.
*Inverse Trig is used to find angles. Keys are above the regular trig functions in blue.
*Pythagorean's Theorem (can only be used for right triangles): a^2 + b^2 = c^2. Used to find missing sides.
*Trigonometry- the study between the relationship between sides and angles of a triangle.
*SOH CAH TOA: the ratios used to find either missing sides or angles in a right triangle.
*Inverse Trig is used to find angles. Keys are above the regular trig functions in blue.
Unit 5: Relations & Functions
12/13- Even & Odd Functions
*Even functions are reflections of each other over the y-axis. Therefore f(x) = f(-x).
*Odd functions are rotations of 180 degrees or reflections over the x and y axes. Therefore -f(x) = f(-x). Odd functions can be flipped upside down and yield the same function.
*Functions can be neither even nor odd.
*Even functions are reflections of each other over the y-axis. Therefore f(x) = f(-x).
*Odd functions are rotations of 180 degrees or reflections over the x and y axes. Therefore -f(x) = f(-x). Odd functions can be flipped upside down and yield the same function.
*Functions can be neither even nor odd.
12/12- Inverse Functions
*Switch x and y and solve for y.
*Inverse functions are reflections of each other over the line y = x.
*A function only has an inverse if it is a 1-1 function.
*You can prove inverse functions by doing a composition of functions (do f of the inverse and the inverse of f of and both should = x).
*Switch x and y and solve for y.
*Inverse functions are reflections of each other over the line y = x.
*A function only has an inverse if it is a 1-1 function.
*You can prove inverse functions by doing a composition of functions (do f of the inverse and the inverse of f of and both should = x).
12/11- Composition of Functions
*Using the output of one function as the input of another function.
*Always start with the inner most function first.
*Read f of g of x.
*2 notations for composition of functions.
*Using the output of one function as the input of another function.
*Always start with the inner most function first.
*Read f of g of x.
*2 notations for composition of functions.
12/11- Functions, Domain and Range
*A function passes the vertical line test and has all unique (different) x values.
*Domain refers to all possible x values for a given function or graph.
*Range refers to all possible y values for a given graph or function.
*Function notation is another way to represent the y- value on a given function or graph. f(x) = y. If they given you a specific x value, simply substitute that value in for all x's and evaluate. This is the y value of the coordinate on the graph.
*A function passes the vertical line test and has all unique (different) x values.
*Domain refers to all possible x values for a given function or graph.
*Range refers to all possible y values for a given graph or function.
*Function notation is another way to represent the y- value on a given function or graph. f(x) = y. If they given you a specific x value, simply substitute that value in for all x's and evaluate. This is the y value of the coordinate on the graph.
Unit 4: Rational Expressions & Equations
11/29 & 11/30- Solving Rational Equations
*Always first determine who "x" can NOT be (will give you undefined fractions).
*To solve a proportion, you cross multiply.
*To solve a rational equation, simply get an LCD for all terms and then remove the denominator. Simplify your numerators and solve.
*Be careful for extraneous roots!!!
*Always first determine who "x" can NOT be (will give you undefined fractions).
*To solve a proportion, you cross multiply.
*To solve a rational equation, simply get an LCD for all terms and then remove the denominator. Simplify your numerators and solve.
*Be careful for extraneous roots!!!
11/26 & 11/28- Adding & Subtracting Rational Expressions
*Find an LCD (Least Common Denominator) by taking all common factors of the denominator as well as those "left over" factors.
*Find new numerators by multiplying.
*Combine numerators, keep the denominator.
*Simplify.
*Find an LCD (Least Common Denominator) by taking all common factors of the denominator as well as those "left over" factors.
*Find new numerators by multiplying.
*Combine numerators, keep the denominator.
*Simplify.
11/16- Roots and Polynomial Functions
*Odd powered functions have a degree that is odd. Even powered functions have degrees that are even.
*Roots are x = #, while factors are (x - #).
*Multiplicity of roots- odd roots CROSS while even roots TOUCH (Think OC and ET).
*Odd powered functions have a degree that is odd. Even powered functions have degrees that are even.
*Roots are x = #, while factors are (x - #).
*Multiplicity of roots- odd roots CROSS while even roots TOUCH (Think OC and ET).
11/15- Synthetic Division
*This process is a little simpler than long division.
*Use only the coefficients when setting up the problem.
*Only multiply and combine rather than determine "how many times does this go into this..."
*If x - a is a factor of the given polynomial, then x = a is a root.
*Remember, roots (when subbed into the original polynomial) should give you 0.
*This process is a little simpler than long division.
*Use only the coefficients when setting up the problem.
*Only multiply and combine rather than determine "how many times does this go into this..."
*If x - a is a factor of the given polynomial, then x = a is a root.
*Remember, roots (when subbed into the original polynomial) should give you 0.
11/14- Long Division
*If (x -a) divides evenly (remainder = 0) into a given polynomial, then x -a is a factor of the given polynomial.
*Long division of polynomials follows the same rules are long division from 4th grade.
*The degree of the quotient will always DESCEND!
*If you end up with 0 at the end, then there is no remainder.
*If (x -a) divides evenly (remainder = 0) into a given polynomial, then x -a is a factor of the given polynomial.
*Long division of polynomials follows the same rules are long division from 4th grade.
*The degree of the quotient will always DESCEND!
*If you end up with 0 at the end, then there is no remainder.
11/13- Simplifying Rational Expressions
*Factor numerator completely
*Factor denominator completely
*Cancel and reduce. Name all values of "x" that make the rational expression undefined
*Factor numerator completely
*Factor denominator completely
*Cancel and reduce. Name all values of "x" that make the rational expression undefined
12/14- Transformations
*If you add to x, then you move to the left.
*If you subtract from x, then you move right.
If you add to y, then you move up.
*If you subtract from y, then you move down.
*If you negate x, then you reflect over the y-axis.
*If you negate y, then you reflect over the x-axis.
*If you multiply f by a whole number it will make it more narrow. Multiply f by a negative whole number and it will be more narrow and a reflection over the x-axis.
*If you multiply f by a fraction, it will become wider. Multiply f by a negative fraction and it will become more wider and a reflection over the x-axis.