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Metric Relations in Right Triangles • By drawing the altitude from the right angle of a right triangle, three similar right triangles are formed Geometric Properties • Using the lengths of the corresponding sides of the triangles formed, we can determine the ratios and from this determine certain geometric properties Property 1 • In a right triangle the length of the leg of a right triangle is the geometric mean between the length of its projection on the hypotenuse A.) • In easiest terms, a leg squared is equal to the hypotenuse multiplied by the leg’s projection on the hypotenuse. Property 2: • The square of the altitude is equal to one part of the hypotenuse multiplied by the other Property 3: • In a right triangle, the product of the length of the hypotenuse and its corresponding altitude is equal to the product of the lengths of the legs. Class Work and Homework • Visions P. 116 # 9 • Visions P. 115 # 2-8, 10, 13, 15, 19 • Math 3000 P. 217 # 1, 3, 4 Trigonometry Definition: • Trigonometry is the art of studying triangles (in particular, but not limited to, right triangles) • Deals with the relationships between the angles and side lengths of a triangle • Before learning the key formulas in trigonometry, it is of absolute importance that some terms are understood • Because we are dealing with right triangles, you are already familiar with one very important right triangle theorem: – The Pythagorean Theorem a² + b² = c² • In every right triangle, because one of the angles measures 90°, then logically the other two angles must add up to 90° A B C Because m<B = 90° then m<A + m<C = 90° (since there are 180° in every triangle) Hypotenuse: – The side that is opposite the right angle – The longest side in the right triangle A Opposite Side: – The side that is opposite of a given angle – Ex: Side AB is opposite m<C Side BC is opposite m<A Adjacent Side: – The side that is neither the hypotenuse or opposite Ex: Side BC is adjacent to m<C Side AB is adjacent to m<A B C Example: Fill in the side that corresponds to the following questions: A Hypotenuse: _________________ Opposite m<A: _________________ Adjacent m<A: _________________ Opposite m<C: __________________ Adjacent m<C: __________________ B C These three definitions of the sides are of utmost importance in trigonometry They are at the root of finding every angle in a right triangle The MOST important Gibberish word you will need to remember in math life SOH – CAH - TOA Opposite sin A Hypotenuse A Adjacent cos A Hypotenuse Opposite tan A Adjacent B C Example: opposite 1 sin 30 hypotenuse 2 adjacent 3 cos 30 hypotenuse 2 opposite 1 tan 30 adjacent 3 3 sin 60 2 1 cos 60 2 3 tan 60 1 A 60° 2 1 30° B C 3 Using Your Calculator 1. The keys sin, cos, tan on the calculator enable you to calculate the value of sin A, cos A, or tan A knowing the measure of angle A So if you know the measure of an angle you can use the sin, cos, or tan buttons on your calculator in order to calculate its value 2. The keys sin-1, cos-1, tan-1 on the calculator enable you to calculate the measure of angle A knowing sin A • So if know sin A, cos A, or tan A, you can calculate the measure of angle A Homework • Hand outs 1 and 2: Trigonometric Ratios and Calculator • Math 3000 pages 228, 229 # 2,3,4,5 P. 182 # 1 P. 184 # 2 Finding Missing Sides Using Trigonometric Ratios Finding the Missing Side of a Right Triangle • In order to find a missing side, you will need two pieces of information: – An angle – One side length In a right triangle 1. Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the hypotenuse, requires the use of sin A Remember: SOH *****Cross Multiply***** Sin 50º= x 5 x=5sin50º = 3.83 cm Finding the measure y of side AC adjacent to the known Angle A, knowing also the measure of the hypotenuse, requires the use of cos A Remember: cos = adjacent/hypotenuse *****Cross Multiply***** y = 5 cos 50º = 3.21 cm Cos 50º = y 5 3. Finding the measure x of side BC opposite to the known angle A, knowing also the measure of the adjacent side to angle A, requires the use of tan A remember tan=opposite/adjacent ***cross multiply*** x tan 30º = 4 x = 4 tan 30º = 2.31 cm Formulas for 90° triangle Formulas to find a missing side Formulas to find a missing angle (hyp)² = (opp)² + (adj)² Sin( A) opp 1 hyp Cos( A) adj 1 hyp Tan( A) opp 1 adj opp A sin 1 hyp adj A cos 1 hyp opp A tan 1 adj Example In the following triangle, find the value of side x (Remember to use SOH-CAH-TOA)!!! x sin 50 5 5 sin 50 x 3.83 x 50 5cm x Example In the following triangle, find the value of side y (Remember to use SOH-CAH-TOA)!!! x cos 50 5 5 cos 50 x 3.21 x 50 5cm x Example • Find the value of x 8 tan 60 x 8 x tan 60 x 4.6 8 x 60 Class work and homework • Finding missing sides • math 3000: page 229 # 6,7 • Hand out Finding Missing Angles using Trigonometry Ratios In a Right Triangle 1. Find the acute angle A when its opposite side and the hypotenuse are known requires the use of sin A SOH – Opposite/hypotenuse sin A = 4 5 M<A=sin-1 ( )=53.1º 4 5 2. Finding the acute angle A when its adjacent side and the hypotenuse are known requires the use of cos A Cos = adjacent/hypotenuse Cos A= 3 4 m<A = cos-1 ( ) = 41.4º 3 4 3. Finding the acute angle A when its opposite side and adjacent side are known requires the use of tan A tan = opposite/adjacent Tan A = 3 2 m<A=tan-1 ( ) = 56.3º 3 2 Find the missing Angle • If we take the inverse of each formula, we can find the missing side angle in a 90° triangle • The symbol for the inverse of sin (A) is sin-1; cos (A) is cos-1; tan (A) is tan-1 Formulas for 90° triangle Formulas to find a missing side Formulas to find a missing angle (hyp)² = (opp)² + (adj)² Sin( A) opp 1 hyp Cos( A) adj 1 hyp Tan( A) opp 1 adj opp A sin 1 hyp adj A cos 1 hyp opp A tan 1 adj Example sin 30º = 0.5 and sin-1 (0.5) = 30º Class work and homework Hand out Find the missing angles using trig ratios Math 3000 page 231 # 8,9 Math 3000 pg 232 # 10-15 10. x=10.4 cm; y=3.0 cm; z=9.0 cm; t=5.2 cm 11. ab=10cos64 = 4.38 cm; bc=10sin64= 8.99 cm; area: 39.4 cm squared 12. H=50tan40 = 42 m 13. Length of shadow = 50tan30 = 28.9 m 14. X=65tan54 degrees = 89.5 m 9. c) d) e) f) B). c=50 degrees, ab=4,6, ac=3.9 mc=60 decrees, ab=5.2, ac=3 Bc=10, angle b=52.1 degrees, angle c=36.9 degrees Ac=12, angle b=67.4, angle c=22.6 Ab=15, angle b=28.1 degrees, angle c=61.9 degrees Solving a triangle To determine the measure of all its sides and angles Sine Law • The sides in a triangle are directly proportional to the sine of the opposite angles to these sides a b c sin A sin B sin C • The sine law can be used to find the measure of a missing side or angle 1st Case • Finding a side when we know two angles and a side We calculate the measure x of AC x 15 15sin 50 x 13.27cm sin 50 sin 60 sin 60 How to: 1. Place Measurement x over sin known angle 2. Equal to 3. Measurement known side over sin of known angle 4. Cross multiply and divide to find unknown measurement 5. Calculate. 2nd Case • Finding an angle when we know two sides and the opposite angle to one of these two sides • We calculate the measure of angle B 10 13 10sin 50 sin B 0.5893 mB 36 sin B sin 50 13 • Make sure you have opposite angles and side measurements. Remember total inside angles must equal 180º How to calculate if need to find an angle: 1. Place side measurement known over sin of angle we wish to know 2. Equal to 3. side measurement over sin angle we know 4. Cross multiply and divide to find x 5. To calculate angle –sin x = angle. Don’t forget unit i.e.º The sine of an obtuse angle • The trigonometric functions (sine, cosine, etc.) are defined in a right triangle in terms of an acute angle. What, then, shall we mean by the sine of an obtuse angle ABC? • The sine of an obtuse angle is defined to be the sine of its supplement. • How to find the measure of the degree of an obtuse angle: • Follow the procedure you have learned so far, then subtract that angle from 180º 10 cm 22º 18.6 cm 10 sin 22 sin w 18.6 18.6sin 22 10 .697 sin .697 44.2 180 44.2 135.5 in Class assignment • Find all missing side lengths and angles and math 3000 page 232 # 10-15 Area of a Triangle There are three formulas to find the area of any triangle • The first of the three you are already aware of: A=bxh 2 Area of a Triangle knowing Two Sides and the Angle in Between Area = ac ● sinB 2 A c Area = ab ● sinC 2 Area = bc ● sinA 2 B b a C • These are the 3 formulas that involve the sine of an angle and the two sides that contain the angle Example: - Find the area of the following triangle 6cm Area = ac●sinB 2 Area = 6 ●12sin(36.3°) 2 Area = 21.3cm² 36.3° 12cm Hero’s Formula Area p( p a)( p b)( p c) Where ‘p’ = HALF of the perimeter Example: p = 6 + 8 + 12 = 26= 13 2 2 8cm 6cm Area 13(13 6)(13 8)(13 12) Area 13(7)(5)(1) Area 455 21.3cm 2 12cm Step 1: Label the sides of the triangle if relevant Step 2: State what information we know Step 3: Select a theorem and find the area • In easiest terms, you need two side lengths and the angle in between them to find the area of any triangle Example: - Find the area of the following triangle 6cm Area = ac●sinB 2 Area = 6 ●12sin(36.3°) 2 Area = 21.3cm² 36.3° 12cm Area of a Triangle Knowing Two Angles and One Side Step 1: Draw the Altitude AD Step 2: Find mAD Step 3: Find mBD 10cm Step 4: Find mDC B A 55 25 D Step 5: Add mBD and mDC Step 6: Use any formula you wish to find the total area C Class work and homework • Math 3000 pg 233 act 1 and 2, and pg 235 numbers 1-5, 7 and hand outs