UNIT 1 Angles 4.1 Sine and cosine 4.2, 4.3, 4.8
HOW TO MEASURE IN BOTH DEGREES ANDRADIANS A degree is a measurement you use in angles. Oneway that you can get an angle is using sine, cosine,or tangent inverse. A radian is when the measure is 1 radian for thecentral angle of a circle if it shows an arc with thesame length like the radius.
KNOW THE CONVERSION BETWEEN DEGREESAND RADIAN MEASURE To convert radians to degrees, you have to multiply180÷π radians Example- 3 π rad × 180° (3/4)×180= 135°4 π rad To convert degrees to radians, you must multiply πradians÷180 Example- 60° × π rad =60 π rad= π/3 rad1 180° 180°
THEIR GRAPHS AS FUNCTIONS AND THECHARACTERISTICS
THEIR DEFINITION AS X- & Y-COORDINATES ONTHE UNIT CIRCLE Sine=y Cosine=x Example: 5π/6 is equal to 150° and the point is(-√3/2,1/2) Sine would equal 1/2 , while cosine equals -√3/2
STUDENTS ARE CAPABLE OF COMPUTING UNKNOWNSIDES OR ANGLES IN A RIGHT TRIANGLE. In order to find a side of a right triangle you can use the PythagoreanTheorem, which is a^2+b^2=c^2. The a and b represent the two shorter sidesand the c represents the longest side which is the hypotenuse. Example- if you have to sides with the measurement of 3 and 5 and youare trying to fine the hypotenuse then you would use the formula3^2+5^2=c^2 . You will get 9+25=c^2 and then 34=c^2 and you wouldhave to square both sides and you get the hypotenuse (c) To get the angle of a right angle you can use sine, cosine, and tangentinverse. They are expressed as tan^(-1) ,cos^(-1) , and sin^(-1) . Example- if you want to fine the angle across from three on the rightangle above then you would use tangent inverse. It would be tan A=3/5and then you multiply both sides by tan^(-1) and you get tan^(-1) 〖(3/5)〗. You put that in the calculator and you get side A.
STUDENTS USE TRIGONOMETRY IN A VARIETYOF WORDS PROBLEMS. Example- “Suppose you are standing on one bank of a river. Atree on the other side of the river is known to be 150 ft. tall. Alone from the top of the tree to the ground at your feet makesan angle of 11° with the ground. How far from you is he baseof the tree?”(x)tan11°=150/x (x).19x/.19=150/.19 150 ftx=789.47 ft. 11° xWord problems are expressed to show real life situations. Wordproblems are usually more difficult than a problem from theactual lesson. The word problems we do are the same as theproblems we usually do but it just requires more thinking.
UNIT 2 Functions of the form f(t)=A sin (Bt + C) &f(t)=A cos (Bt + C): 4.4, 4.5, 4.7, 4.8
PROPERTIES: AMPLITUDE, FREQUENCY,PERIOD AND PHASE SHIFT (A, B & C)
STUDENT WILL BE ABLE TO TAKE A GIVEN ANGLE AND COMPUTE THETRIGONOMETRIC FUNCTION AND ITS INVERSE WITH THE AID OF THE UNITCIRCLE (BY HAND)
STUDENTS USE TRIGONOMETRY IN A VARIETYOF WORD PROBLEMS. “When sitting atop a tree and looking down at his pal Joey, the angleof depression of Mack’s line of sight is 38°32’. If joey is know to bestanding 39 feet from the base of the tree, how tall is the tree ?h39 ft 38°32’ First you must convert the 32 into degrees so you divideit by 60 and then add it to 38 and you get 38.5°. Then to get the height you have to use tangent and youuse tan38.5°=h/39 and you get the height of the tree tobe approximately 31 ft.
VECTORS-ADDITION AND SCALARMULTIPLICATION U=(u1,u2), v=(v1,v2) are vectors. When you are adding youuse the formula u+v=(u1+v1,u2,v2). K is known to be a realnumber and you would use it to multiply with the formulaku=k(u1,u2)=<ku1,ku2>. Example- let u=<-1,3> and v=<4,7> and add the vectors. u+v=(-1,3)+(4,7)=(-1+4,3+7) <-3,10> All you have to do is add the first number from each pointto get the x and add the second number from each pointtogether to get the y. Example- use scalar multiplication to find 3u whenu=<-1,3>. Since there is a 3 before the u, you have to multiply eachnumber in the point by three. 3u=3(-1,3)=(-3,9)
VECTORS-RESOLVING THE VECTOR The formula v=(|v|cosƟ,|v|sinƟ) can be used when v has adirection angle Ɵ and the components of v can be calculated.The unit vector in the direction of v is u=v/|v|=(cosƟ,sinƟ). Example- “find the components of vector v with direction angle115 and magnitude 6.” All you have to do is substitute 6 which is the magnitudeinto the equation wherever there is a v. Also plug in 115wherever there is a Ɵ. V=(a,b)=(6cos115 ,6sin115 ) a would be approximately -2.54 while b would be about 5.44