Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values Aim: demonstrate how basic values of the trig functions are found 1 1

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values We shall demonstrate how to find basic trig values with a few values from the first quadrant. Values in other quadrants will be built up from these, as shown later. The most basic values are those shown below. DegreesRadians 00 30°π/6 45 °π /4 60 °π /3 90 °π /2

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values  =0, π/2 Note the position of points P, Q: (1,0), (0,1) 1 1 P Q

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values The coordinates of P tell us that cos(0) = 1, sin(0)=0 and tan(0)=0/1=0. From the coordinates of Q we find cos(π/2)=0, sin(π/2)=1, tan(π/2)=1/0, ie undefined.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values The first two values were very easy, and indeed it is extremely simple to find the trig functions for all multiples of π/2.  PCosSinTan π(-1,0)00 3 π/2(0,-1)0DNE 2 π(1,0)100

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values This is all very well for simple multiples of π/2, but what about other angles? Let us try several simple angles to illustrate the sorts of approaches. Try the angle  = π/4. Now  = π/4 leads to the following diagram, where the values of x and y, ie cos(  ) and sin(  ), are still unknown. However we can say that x and y must be equal, and so the triangle formed must be isosceles.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values 1 1 x y π/4 P

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values Applying Pythagoras Theorem’ to the triangle below, knowing that x and y are equal, we find: x 2 +y 2 =1 2x 2 =1 x 2 =1/2 x=1/√2 Hence cos(π/4)=sin(π/4)=1/√2 Of course tan(π/4)=1. Note that some texts will write 1/√2 as √2/2 y 1 x

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values Let us now find the trig functions of 60 ° (π /3) and 30° (π/6). We shall use the trig relations (known as identities) sin(  )=cos(90-  ), cos(  )=sin(90-  ), tan(  )=cot(90-  ) Construct first an triangle with angle θ= 60 ° and sides OP and OQ both of length 1. OP and OQ are radii, so they must both be of length 1. (See following slide). We leave the unit circle in place. Triangle POQ is isoceles, so angles OPQ and OQP must be equal. In fact both are (180-60)/2= 60 °. So the triangle is an equilateral triangle and hence all sides are 1 and all angles are 60 °.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values Add an extra line vertically from the top to the x axis as in the previous diagram. This splits the bottom side evenly in two equal halves. Note that because the triangle is constructed neatly within the unit circle it must have sides of length 1, and the constraint that the triangle is equilateral means that every side is of length 1. The cosine and sine of 60 ° will then be x and y in the preceding diagram.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values It is now clear that cos(60 ° )=1/2, as 2x=1. As a result, (1/2 )2 +y 2 =1 or y=sin(60 ° )= √3/2. Also tan(60 ° )=√3. Using the fact that sin(  )=cos(90-  ), we find sin(30 °)=cos(60 °)=1/2. Using cos(  )=sin(90-  ), we find cos(30 °)=sin(60 ° )=√3/2. So tan(30 °)=1/√3.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values Hence our table of basic trig functions is as on the next slide. It is imperative that this table be memorised. Slide 16 gives some hints to help with this.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values  sincosTan 0010 π/61/2√3/21/ √3 π/41/ √2 1 π/3 √3/2 1/2√3 π/210DNE

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Basic Trig Values Firstly note that all multiples of π/2 can easily be found by drawing the unit circle and remembering that the circle cuts the axes at ±1. Secondly, remember that sin and cos of π/4 are equal and are both 1/ √2. the tan must then be 1. The values for π/6 and π/3 come from 1/2 and √3/2. Which is which? When  =0, the circle is nearly vertical and is climbing quite steeply. Hence it will reach the halfway mark (ie y=1/2) quite early on, ie before  = π/4. Hence sin(π/6)=1/2 and the other values follow.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Inverse Trig Functions Note that we often need to know the inverse trig functions, such as sin -1 (x), cos -1 (x) and tan -1 (x). These are not powers of the trig functions. We write positive powers as simple exponents, eg sin 2 x, cos 3 x, etc. But negative powers need to be handled differently. Eg 1/tan(x)=(tan(x)) -1, not tan -1 (x). We reserve the notations tan -1 (x), sin -1 (x) and cos -1 (x) for the inverse functions. For example, since sin(x) gives us the sin of angle x, sin -1 (x) gives us the angle whose sin is x. Likewise as cos(x) gives us the cos of angle x, cos -1 (x) gives us the angle whose cosine is x.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics Finding Inverse values This is just a matter of reading off the correct values from the table. Eg, as sin(  /6)=1/2, sin -1 (1/2)=  /6. Since tan(  /4)=1, tan -1 (1)=  /4. Since cos(  /6)=  3/2, cos -1 (  3/2)=  /6.

Educating Professionals – Creating and Applying Knowledge - Serving the Community University of South Australia School of Mathematics The Washup Memorising the above table and using the unit circle concept we can find the trig functions a very wide range of angles. The circle concept and table will also allow us to find inverse trig functions as well. The next slide show, Extending the Domain, shows how to extend the domain.