1、Unit 3: Trigonometry,Lesson 1: Review of Trig & Radian Measure,Trigonometry Review,The Primary Trigonometric Ratios are: is the Greek letter “theta” and is the common symbol for an angleWe use SOHCAHTOA to memorize these ratios,S,O,H,C,A,H,T,O,A,The Reciprocal Trig Ratios,Recall from Unit 2: Recipro
2、cal means you put “one over” or, more simply, you “flip it”Lets flip the primary trig ratios:,sin ,=,opposite,hypotenuse,1,cos ,=,adjacent,hypotenuse,1,tan ,=,opposite,adjacent,1,The Reciprocal Trig Ratios,These ratios have special names and short forms: cosecant csc secant sec cotangent cotAnd they
3、 are equal to:,Example 1,Determine the reciprocal trig ratios of the 30- 60 special triangle,Example 1: Solution,The 30- 60 special triangle looks like:The primary ratios are therefore:,We now flip these to get the reciprocal ratios,Example 2,(a) Determine the value of csc 25 to 4 decimal places(b)
4、Determine the value of (to the nearest degree) given that sec = 2.34,Example 2: Solution,Before proceeding, make sure your calculator is in DEGREES On your TI-83 press MODE, move down to DEGREES and press ENTER. Then 2nd QUIT(a)Because there is no “csc” button on your calculator, we need to use the
5、definition of cosecant:,There is a “sin” button on your calculator,Example 2: Solution,(b) Because there is no “sec” button on your calculator, we need to use the definition of secant:,Flip both sides to get cos up top,Remember to use inverse trig when you NEED an angle,Radians,At the start of Examp
6、le 2 I told you to make sure your calculator was in DEGREESThe reason for this is that there is another unit used to measure angles the RADIAN The default unit for angles on a TI-83 is the radian, not the degree If you need to make a change: press MODE, move down to RADIANS and press ENTER. Then 2nd
7、 QUITTo understand where the radian came from, lets look at the unit circle,Understanding Radians,Draw the unit circle:,UNIT CIRCLE Radius = 1 unit Centre at origin,The circumference of this circle is:,An angle representing one complete revolution of the unit circle is 360,C = 2r = 2(1) = 2,The radi
8、an is defined such that these two quantities are equal:,2 radians = 360,Or, more commonly, radians = 180,Converting Between Radians & Degrees,#1 To change radians to degrees, multiply by,. #2 To change degrees to radians, multiply by,.,Given that 180 = radians:,Example 3,Complete the following conve
9、rsions. Give exact and approximate values.(a) 30 (a) 270,Example 3: Solution,30,Exact value:,(a),Approximate:,270,Exact value:,(b),Approximate:,Example 3: Notes,When writing angles in radians there are three options for specifying the unit: Writing the unit name 1.25 radiansUsing the short form “rad
10、” 1.25 radUsing no unit at all 1.25,Example 4,Convert the following radian measures to degrees,(b) 5.86,Example 4: Solution,(a),5.86,(b),Why Radians?,Without radians it would not be possible to calculate arc length the distance travelled around part of a circle The formula for calculating arc length
11、 isa is the arc length r is the radius of the circle is the angle of revolution in radians,Example 5,How far does a car travel around a circular corner if it moves through an angle of 25 and the circular corner has a radius of 150 m?,Example 5: Solution,25,We first need to convert 25 to radians,No w
12、e apply use arc length formula:,The car travelled 65.4 m,Summary, Part 1,The Primary trig ratios are:The reciprocal trig ratios are:,cosecant,secant,cotangent,Summary, Part 2,The radian is an alternate unit for measuring angles. 180 = radians To change radians to degrees, multiply by To change degrees to radians, multiply bySometimes, no unit symbol is given for angles measured in radiansWith radians we can calculate arc length Use a = r, where is in radians,Practice Problems,P. 200 #6, 7P. 208 #1-9,
