7In the ratios: x is an angle (not the 90 degree angle) “adjacent”, “opposite” and “hypotenuse” are all side lengths, not angles“Adjacent” is the side next to the knownangle“Opposite” is the side across from theknown angle
8!IMPORTANT! To solve trig functions in the calculator, make sure to set your MODE to DEGREESDirections:Press MODE, arrow down to RadianArrow over to DegreesPress ENTER
10Measuring AnglesThe measure of an angle is determined by the amount of rotation from the initial side to the terminal side.There are two common ways to measure angles, in degrees and in radians.We’ll start with degrees, denoted by the symbol º.One degree (1º) is equivalent to a rotation of of one revolution.
19Graphs of sine & cosine Fundamental period of sine and cosine is 2π. Domain of sine and cosine isRange of sine and cosine is [–|A|+D, |A|+D].The amplitude of a sine and cosine graph is |A|.The vertical shift or average value of sine and cosine graph is D.The period of sine and cosine graph isThe phase shift or horizontal shift is
20Sine graphs y = sin(x) y = sin(x) + 3 y = 3sin(x) y = sin(3x)
21Graphs of cosine y = cos(x) y = cos(x) + 3 y = 3cos(x) y = cos(3x)
22Tangent and cotangent graphs Fundamental period of tangent and cotangent is π.Domain of tangent is where n is an integer.Domain of cotangent where n is an integer.Range of tangent and cotangent isThe period of tangent or cotangent graph is
23Graphs of tangent and cotangent y = tan(x)Vertical asymptotes aty = cot(x)Vertical asymptotes at
24Graphs of secant and cosecant y = csc(x)Vertical asymptotes atRange: (–∞, –1] U [1, ∞)y = sin(x)y = sec(x)Vertical asymptotes atRange: (–∞, –1] U [1, ∞)y = cos(x)
26Do you remember the Unit Circle? Where did our Pythagorean identities come from??Do you remember the Unit Circle?What is the equation for the unit circle?x2 + y2 = 1What does x = ? What does y = ?(in terms of trig functions)sin2θ + cos2θ = 1Pythagorean Identity!
27Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos2θsin2θ + cos2θ =cos2θ cos2θ cos2θtan2θ = sec2θQuotientIdentityReciprocalIdentityanother Pythagorean Identity
28Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin2θsin2θ + cos2θ =sin2θ sin2θ sin2θcot2θ = csc2θQuotientIdentityReciprocalIdentitya third Pythagorean Identity
34Types of AnglesThere are four different types of angles in any given circle. The type of angle is determined by the location of the angles vertex.1. In the Center of the Circle: Central Angle2. On the Circle: Inscribed Angle3. In the Circle: Interior Angle4. Outside the Circle: Exterior Angle* The measure of each angle is determined by the Intercepted Arc
35Intercepted ArcIntercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds:1. The endpoints of the arc lie on the angle.2. All points of the arc, except the endpoints, are in the interior of the angle.3. Each side of the angle contains an endpoint of the arc.
36Central AngleDefinition: An angle whose vertex lies on the center of the circle.Central Angle(of a circle)Central Angle(of a circle)NOT A Central Angle(of a circle)* The measure of a central angle is equal to the measure of the intercepted arc.
37Measuring a Central Angle The measure of a central angle is equal to the measure of its intercepted arc.
38Inscribed AngleInscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).Examples:3124Yes!No!Yes!No!
39Measuring an Inscribed Angle The measure of an inscribed angle is equal to half the measure of its intercepted arc.
40CorollariesIf two inscribed angles intercept the same arc, then the angles are congruent.
41An angle inscribed in a semicircle is a right angle. Corollary #2An angle inscribed in a semicircle is a right angle.
42Corollary #3If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.** Note: All of the Inscribed Arcswill add up to 360
44Area of a Sector Formula measure of the central angle or arcmπr2Area of a sector =360The area of the entire circle!The fraction of the circle!.
456. The area of sector AOB is 48π and . Find the radius of ○O.mπr2Area of a sector =36027048π =πr236041634r248 =343r264 =r= 8
467. The area of sector AOB is and. Find the radius of ○O.mπr2Area of a sector =360940π =πr243609919=r2149181=r249r =2
47The standard form of the equation of a circle with its center at the origin is r is the radius of the circle so if we take the square root of the right hand side, we'll know how big the radius is.Notice that both the x and y terms are squared. Linear equations don’t have either the x or y terms squared. Parabolas have only the x term was squared (or only the y term, but NOT both).
48Let's look at the equation This is r2 so r = 3The center of the circle is at the origin and the radius is 3. Let's graph this circle.2-7-6-5-4-3-2-11573468Count out 3 in all directions since that is the radiusCenter at (0, 0)
49The center of the circle is at (h, k). This is r2 so r = 4 If the center of the circle is NOT at the origin then the equation for the standard form of a circle looks like this:The center of the circle is at (h, k).This is r2 so r = 4Find the center and radius and graph this circle.The center of the circle is at (h, k) which is (3,1).2-7-6-5-4-3-2-11573468The radius is 4
50If you take the equation of a circle in standard form for example: This is r2 so r = 2(x - (-2))Remember center is at (h, k) with (x - h) and (y - k) since the x is plus something and not minus, (x + 2) can be written as (x - (-2))You can find the center and radius easily. The center is at (-2, 4) and the radius is 2.But what if it was not in standard form but multiplied out (FOILED)Moving everything to one side in descending order and combining like terms we'd have:
514 16 4 16 Move constant to the other side If we'd have started with it like this, we'd have to complete the square on both the x's and y's to get in standard form.Move constant to the other sideGroup x terms and a place to complete the squareGroup y terms and a place to complete the square416416Complete the squareWrite factored and wahlah! back in standard form.