What is Trigonometry? The study of three sided polygons!

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1 What is Trigonometry? The study of three sided polygons!
Three sided measure! Sounds like a triangle to me! Intro

2 Why study Trig? If you understand the triangle you understand every shape in the universe. Every shape is made of triangles. why study

3 review of some geometry
Some basic Geometry refresher you will need: Opposite angles are congruent Opposite angles are congruent Congruent means ‘equal’ but for shapes review of some geometry

4 90° 180° Measuring an angle around a point on a line to the opposite direction is a 180° angle 180 angle

5 Angle AZB is congruent to Angle PZQ; or
AZB  PZQ AZB PZQ angle

6 Basic Geometry: Transversals
This symbol: Parallel Lines on a pair of lines shows they are going the same direction; ie: they are parallel A line that cuts a pair of other lines is called a transversal The green and red markers show congruent angles transversals

7 Similar triangles Similar triangles are the same shape but not necessarily the same size. That means all three angles in the one match up or ‘correspond’ with the other. To show which sides ‘correspond’ (or go together) we mark them with pairs of similar marks. similar triangles

8 To say these triangles are similar we say:
ABC~ PQZ notice this big triangle is horizontally flipped relative to the little one P A Q B C Z sim triangles 2

9 Did you know? All three corner angles of a triangle add up to 180°
B A + B + C = 180° 70° 60° A C Given the measure of A and C; what is the measure of angle B? mB = ________ ? 50° sum=180

10 Triangle ABC and triangle APQ are similar ie: ABC~ APQ
similar triangles

11 A B C P Q 5 7 10 4 Mommy Triangle: ABC Baby Triangle: APQ
Length AP of the baby triangle corresponds with length AB of the mommy triangle. The mommy is twice as big as the baby, so all of the mommy’s sides are twice as big as the baby’s! BC = 2 * 4 = 8 AC = 2 * 7 = 14 sim triangle 2

12 So: A B C P Q 5 7 10 4 so AC = 14 and BC = 8 Or more correctly:
solve sim triangle

13 A B C P Q 8 12 9 24 Practice: Find side AC and side BC AC = 36 BC = 27
Caution: clearly these triangles are not properly ‘drawn to scale’. There is no way if you measure it that AB would be 3 times as long as AP. But go by what the numbers say! Not by the drawing! solve sim triangle 2

14 D Find CZ and BZ 25 20 Z 6 A C 8 B They are similar triangles:
AZB = CZD since they are ‘opposite angles’ BAZ and DCZ are both 90°. So the remaining corresponding angles must be congruent. Find CZ and BZ So mom is 2.5 times bigger than baby So CZ = 15 Did you notice you could have used Pythagoras also to solve for the two sides? So BZ = 10 solve sim triangle 3

15 Any triangle can be made up of two right-angle triangles.
We will limit further discussion of triangles to simple right-angle triangles. But any triangle can be made up of two right-angle triangles, so we are not really compromising our studies of triangles by only examining right-angle triangles. Right angle triangle. Any triangle can be made up of two right-angle triangles. right angle triangles

16 naming parts of the triangle
Naming parts of the right triangle The hypotenuse is always the side across from the right angle. It is always the longest side. Hypotenuse leg leg naming parts of the triangle

17 labelling the triangle
Labelling triangles Here is another way to label parts of a triangle for a simple single triangle. Corners are labelled with capital letters. Sides are labelled with little letters of the corner across from them. A b c B a C labelling the triangle

18 Sometimes instead of just talking about the measure of angle R (mR) we give it an even easier symbol. The most common symbol is the Greek letter ‘theta’; . Did you know? there are lots of Greek letters, you have probably used a few before like: , , , etc. P p Q q R r  angles and sides

19 When we are interested in a particular angle in a right-angle triangle we name the parts of the triangle as below. P p Q q R r  Hypotenuse Opposite Side Adjacent side naming sides

20 animated gif

21 We already agree from similar triangles that:
and that all the corners have the same corresponding angles including the value of  A D C B E  cross multiplying gives us: Opposite Adjacent So the ratio, or fraction of, opposite side of  divided by the adjacent side of  is the same for all triangles with the same angle  tan ratio

22 Tangent ratio Opposite  Adjacent
We call the ratio of the side opposite the angle  to the side adjacent to the angle the ‘tangent of the angle’, or tan(). This triangle has a tangent of The opposite side of the triangle is 67% as long as the adjacent side. Every triangle in the universe like this has an angle  of 33.8  Adjacent Opposite 33.8° 6.7 10 tangent

23 Tangent ratios Over the last three thousand years we have measured the angle that goes with each tangent ratio! Here are a selected few of the results: Opp 0.577 1 30 60 17.32 10 Adj 10 45 Opp Adj tangent ratios

24 know the tan no the shape
Know the tangent know the shape! So now when someone tells you the tangent of an angle in a right– angle triangle you can instantly draw the shape of the triangle!! So: Draw me a right-triangle that has a corner angle whose tangent is 2! Easy! The Opposite side from the corner is just twice the Adjacent side! 3 6  Opp Adj It turns out after 3000 years of measuring that every right-angle triangle with a tan of 2 has a corner angle of 63.4°. know the tan no the shape

25 In the case below the sin of  is 0.5
Sine ratio The sine ratio is the comparative ratio of the Opposite and Hypotenuse In the case below the sin of  is 0.5 A D C B E  Opposite Hypotenuse It turns out that every triangle in the universe with a sine of 0.5 has exactly a 30° angle. 5 10 sine

26 Sine ratios Over the last three thousand years we have measured the angle that goes with each sine ratio! Here are a selected few of the results: 0.5 Opp 1 30 60 1.732 2 7.07 10 45 Opp sine ratios

27 Animation of Similar Triangles and Trig Ratios

28 Cosine ratio The idea of cosine is the same as the other ratios except we compare the Adjacent and the Hypotenuse sides It turns out that every right-triangle with a cosine ratio of is a 30° angle.  10 8.66 cosine

29 SOH CAH TOA The three ratios we have learned to give the shape of a right-angle triangle are: Most students memorize this as: SOH CAH TOA Write it at the top of every page of trig problems or tests In theory you only really need one of these trig ratios, we have several just for convenience. In grade 12 you will learn three more trig ratios! Six ways to describe the shape of a triangle! SOH CAH TOA

30 Solve triangles – find side
Find side x 10 x 30° We use the Tan ratio since we do not readily know the Hypotenuse used by the other two ratios 5.77 but we know from the tangent table or a calculator that tan30° = 0.577 So x = 10*0.577 or 5.77 solve triangles

31 Solve triangles – find side
Find side x We use the Sin ratio since we do not readily know the Adjacent used by the other two ratios x 10 15° but we know from the sine table or a calculator that sin15° = 0.259 So x = 10*0.259 or 2.59 solve triangles 2

32 What angle has a cosine that is 0.259?
Solve for an angle Find angle  We only know three formulas:  SOH CAH TOA but we are given the Adjacent side and the Hypotenuse side, so using Cosine would be easiest. 20 Hypotenuse What angle has a cosine that is 0.259? Adj Angle  = 75° 5.18 solve for an angle

33 Lots of patterns here, can you see them?
Trig Table Lots of patterns here, can you see them? trig table