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Module 1: Angles in a Unit Circle
Angles - formed by rotating a ray about its endpoint.
Positive Angles - angles formed by a counter clockwise direction.
Negative Angles - angles formed by a clockwise direction.
Angle in Standard Position - an angle is in standard position if it is drawn in the Rectangular Coordinate System with its vertex at the origin and its initial side on the positive x-axis.
● Straight angle – is an angle of 180°
● Right angle – is an angle of 90°
● Acute angle – is an angle whose measure is between 0° and 90°
● Obtuse angle – is an angle whose measure is between 90° and 180°
● Reflex angle – is an angle whose measure is between 180° and 360°
Quadrantal Angles - an angle in standard position whose terminal side lies on the x – axis or y – axis.
Complementary Angles – angles that add up to 90°
Supplementary Angles - angles that add up to 180°
Unit Measure of Angle
Radian Measure:
● Radian (rad) – is the measure of the central angle subtended by an arc of a circle that is equal to the radius of the circle. One radian is about 53.7°
● Degree – is the measure of an angle formed by rotating a ray 1 360 of a complete revolution.
Conversion between Degrees to Radians
● To convert degrees to radians, multiply degrees by (π radian)/(180°)
● To convert radians to degrees, multiply radians by (180°)/(π radian)
Module 2: Standard Angle and Coterminal Angle
Standard Angle - an angle is in standard position if its vertex is located at the origin and one ray is on the positive x-axis.
● The ray on the x-axis is called the initial side and the other ray is called the terminal side.
Coterminal Angle – an angle in standard position (angles with the initial side on the positive x -axis) that have a common terminal side.
Reference Angle - each angle drawn in standard position, there is a related angle called reference angle.
● The reference angle is the acute angle (the smallest angle) formed by the terminal side of the given angle and the x-axis.
● Reference angles may appear in all four quadrants.
● Angles in quadrant I are their own reference angles.
0° ≤ Ө ≤ 90° = 𝜽 ㅤㅤㅤㅤㅤㅤㅤㅤ90° ≤ Ө ≤ 180° = 𝟏𝟖𝟎° – 𝜽
180° ≤ Ө ≤ 270° = 𝜽 – 𝟏𝟖𝟎° ㅤㅤㅤ270° ≤ Ө ≤ 360° = 𝟑𝟔𝟎° − 𝜽
Module 3: Circular Functions
● “Circular Functions” are named the same as trig functions (sine, cosine, tangent, etc.)
● The domain of trig functions is a set of angles measured either in degrees or radians
● The domain of circular functions is a set of real numbers
Unit Circle - a circle whose center is at the origin and with a radius of 1 unit.
Some of the points on the unit circle are: (1, 0), (0, 1), (-1, 0) and (0, -1)
Equation of a Circle: (x – h)² + (y – k)² = r²
(h, k) is the coordinate of the center of the circler is the radius(x, y) is the point of the circle
Trigonometric Functions using Unit Circle
sinθ = y cscθ = 1/y
cosθ = x secθ = 1/x
tanθ = y/x cotθ = x/y
Signs of Six Trigonometric Functions in each Quadrants
Quadrant I:
positive = all
Quadrant II:
positive = sinθ & cscθ
negative = cosθ, secθ, tanθ, & cotθ
Quadrant III:
positive = tanθ & cotθ
negative = sinθ, cscθ, cosθ, & secθ
Quadrant IV:
positive = cosθ & secθ
negative = sinθ, cscθ, tanθ, & cotθ
Finding Trigonometric Function Values of Angles Measured in Radians
● All previous definitions of trig functions still apply.
● Sometimes it may be useful when trying to find a trig function of an angle measured in radians to first convert the radian measure to degrees.
● When a trig function of a specific angle measure is indicated, but no units are specified on the angle measure, ALWAYS ASSUME THAT UNSPECIFIED ANGLE UNITS ARE RADIANS.
● When using a calculator to find trig functions of angles measured in radians, be sure to first set the calculator to “radian mode”.
Module 4: Graphs of Circular Functions
Periodic Function
A function f is periodic id there exist a positive real number t such that f (s + t) = f(s) for all s in the domain of f. The smallest number t is periodic is called the period f.
Periodic Properties:
sin (s + 2𝜋) = sin s csc (s + 2𝜋) = csc s
cos (s + 2𝜋) = cos s sec (s + 2𝜋) = sec s
tan (s + 2𝜋) = tan s cot (s + 2𝜋) = cot s
Even – Odd Properties
The cosine ad secant function are even.
cos (-s) = cos s ㅤㅤsec(-s) = sec s
The sine, cosecant, tangent and cotangent are odd.
sin (-s) = -sin s ㅤㅤtan (-s) = -tan s
csc (-s) = -csc s ㅤㅤcot (-s) = -cot s
Properties of Sine and Cosine Functions
The graphs of y = sin x and y = cos x have similar properties:
1. The domain is the set of real numbers.
2. The range is the set of y values such that −1 ≤ 𝑦 ≤ 1.
3. The maximum value is 1 and the minimum value is –1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the range over an x-interval of 2𝜋.
6. The cycle repeats itself indefinitely in both directions of the x-axis.
A. Amplitude of Sine and Cosine Curves
The amplitude of a y = a sin x and y = a cos x represents half the distance between the maximum and minimum values of the function and is given by Amplitude = ǀ a ǀ. Amplitude also indicates the height of the graph.
B. Period of Sine and Cosine Functions
The period is the length of one complete cycle. Let b be a positive real number. The period of y = a sin bx and y = a cos bx is given by Period p = 2𝜋/𝑏.
C. Graphs of Sine and Cosine Functions
Phase Shift - it is the amount by which the graph is shifted to the right or to the left based from the base curve.
D. Vertical Translations
Vertical Translations - is the upward or downward movement of the graph.
Module 5: Problems Involving Circular Functions
Arc Length and Central Angles
The radian measure 𝜽 of a rotation is the ratio of the distance s traveled by a point at the radius r from the center of rotation, to the length of the radius r
When using the formula 𝜃 = s/r , 𝜃 must be in radian and s and r must be expressed in the same unit.
Linear Speed - defined to be distance traveled per unit of time. If we use v for linear speed, s for distance, and t for time, linear speed is defined as v = s/r
Angular Speed - defined to be amount of rotation per unit of time, the Greek letter 𝜔 (omega) is generally used for angular speed. Thus, for a rotation 𝜃 and time t, angular speed is defined as 𝜔 = θ/t
Linear Speed in Terms of Angular Speed
The linear speed 𝝂 of a point a distance r from the center of the rotation id given by, 𝝂 = 𝒓𝝎, where 𝜔 is the angular speed in radians per unit time.
For the formula 𝜈 = 𝑟𝜔, the units of distance for v and r must be the same, 𝜔 must be in radians per unit of time, and the units of time for v and 𝜔 must be the same.
Area of a Circular Sector
The area of a circle of radius r is A = 𝜋𝑟². A sector of this circle with central angle 𝜃 has an area that is the fraction θ/2π of the area of the entire circle.
Problem Solving Process:
1. Understand the problem (Analysis)
2. Devise a plan (Planning)
3. Carry out the plan (Implementation)
4. Look back (Reflection)
Module 6: Trigonometric Identities
Trigonometric Identities - a statement of equality that is true for all values where the function is defined. The equation in reciprocal, quotient, and Pythagorean identities which follow each solution is true for all values of 𝜃 for which both sides are defined.
Reciprocal Identities:
sin θ = 1/cscθ ㅤㅤcsc θ = 1/sinθ
cos θ = 1/secθㅤㅤsec θ = 1/cosθ
tan θ = 1/cotθㅤㅤcot θ = 1/tanθ
Quotient Identities:
tan θ = sinθ/cosθ ㅤㅤcot θ = cosθ/sinθ
Pythagorean Identities:
sin²θ + cos² = 1 ㅤㅤtan²θ + 1 = sec²θ ㅤㅤ1 + cot² = csc²θ
Identity Equation - an equation that is true for all valid replacement of the variable.
Conditional Equation - is a statement that is true on condition that the variable is replaced with the correct value.
● Finding Trigonometric values using trigonometric identities
● Simplifying Trigonometric Identities
● Proving Trigonometric Identities
Guidelines for Verifying Identities:
1. Start with the most complicated side of the equation.
2. Factor an expression, add fractions, square a binomial, or create s monomial denominator, if possible.
3. Use the fundamental identities, whenever possible.
4. Convert all terms to sines and cosines.
5. Always try something.
Module 7: Sum and Difference of Two Angles and Other Identities
● Proving Trigonometric Identities
● Evaluating Trigonometric Functions
● Proving Double – Angle Identities
● Evaluating Trigonometric Functions involving Double – Angle Identities
Formulas:
The General Addition Formulas
1. cos (𝛼 + 𝛽) = cos 𝛼 cos 𝛽 – sin 𝛼 sin 𝛽
2. cos (𝛼 - 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽
3. sin (𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽
4. sin (𝛼 - 𝛽) = sin 𝛼 cos 𝛽 – cos 𝛼 sin 𝛽
5. tan (𝛼 + 𝛽) = (tan α + tan β )/(1- tanα tan β)
6. tan (𝛼 - 𝛽) = (tan α - tan β )/(1- tanα tan β)
Double-Angle Formulas
1. sin 2𝜃 = 2 sin 𝜃 cos 𝜃
2. cos 2 𝜃 = 𝑐𝑜𝑠2𝜃 - 𝑠𝑖𝑛2𝜃
ㅤㅤㅤㅤ= 2 𝑐𝑜𝑠²𝜃 – 1
ㅤㅤㅤㅤ= 1 - 2 𝑠𝑖𝑛²𝜃
3. tan 𝜃 = (2tanθ )/(1- tan^2 θ)
Half-Angle Formulas
1. cos θ/2 = ± √((1+cosθ)/2)
2. sin θ/2 = ± √((1-cosθ)/2)
3. tan θ/2 = ± sinθ/(1-cosθ)
ㅤㅤㅤㅤ= (1-cosθ)/sinθ
Module 8: Inverse Trigonometric Functions and Trigonometric Equations
Notation:
Notation is used because in the language of composition of functions, we can write: f o f-1 = I → This is similar in form to the multiplication of numbers, a· a-1 = 1
Existence of an Inverse Function:
A one-to-one function, is a function in which for every x there is exactly one y and for every y, there is exactly one x. A one-to-one function has an inverse that is also a function.
● If a function passes both the vertical line test (so that it is a function in the first place) and the horizontal line test (so that its inverse is a function), then the function is one to-one and has an inverse function.
● Understanding and Using the Inverse Sine, Cosine, and Tangent Functions
● Determining the principal values of the inverse trigonometric functions
● Graphing: Inverse Function
Trigonometric Equations
Trigonometric Equations - is a conditional equation that involves trigonometric functions.
● In solving trigonometric equations, it is enough to find solutions from 0 to 2𝜋.
Steps in Solving Trigonometric Equations
1. If the equation is linear in one trigonometric function:
a. Directly solve for the trigonometric function.
b. Solve for the angle by determining the function values of the quadrantal or special angle by applying the trigonometric tables.
2. If there are more trigonometric functions, apply the Fundamental Identities to represent the equation in terms of one trigonometric function.
3. If the equation is not linear, represent it by isolating the left side of the equation. Then apply factoring (if factorable), otherwise use quadratic formula.
4. Use algebraic techniques for solving the trigonometric equations.