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° Because m<B = 90° then m<A + m<C = 90° (since there are 180° in every triangle)

Hypotenuse: Opposite Side: Adjacent Side: The side that is opposite the right angle The longest side in the right triangle 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

Example: Fill in the side that corresponds to the following questions: Hypotenuse: _________________ Opposite m<A: _________________ Adjacent m<A: _________________ Opposite m<C: __________________ Adjacent m<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

Example: 60° 2 1 30°

Using Your Calculator 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

Hand outs 1 and 2: Trigonometric Ratios and Calculator 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 Sin 50º= x=5sin50º = 3.83 cm 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=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***** Cos 50º = y = 5 cos 50º = 3.21 cm

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*** tan 30º = 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

Example In the following triangle, find the value of side x (Remember to use SOH-CAH-TOA)!!!

Example In the following triangle, find the value of side y (Remember to use SOH-CAH-TOA)!!!

Example Find the value of x

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 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 = M<A=sin-1 ( )=53.1º

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= m<A = cos-1 ( ) = 41.4º

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 = m<A=tan-1 ( ) = 56.3º

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

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 x=10.4 cm; y=3.0 cm; z=9.0 cm; t=5.2 cm ab=10cos64 = 4.38 cm; bc=10sin64= 8.99 cm; area: 39.4 cm squared H=50tan40 = 42 m Length of shadow = 50tan30 = 28.9 m X=65tan54 degrees = 89.5 m 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

To determine the measure of all its sides and angles 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

The sine law can be used to find the measure of a missing side or angle

1st Case We calculate the measure x of AC Finding a side when we know two angles and a side We calculate the measure x of AC

How to: Place Measurement x over sin known angle Equal to Measurement known side over sin of known angle Cross multiply and divide to find unknown measurement 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

Make sure you have opposite angles and side measurements 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: Place side measurement known over sin of angle we wish to know Equal to side measurement over sin angle we know Cross multiply and divide to find x 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

Class assignment Find all missing side lengths and angles and math 3000 page 232 # 10-15

There are three formulas to find the area of any triangle 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 = b x h 2

Area of a Triangle knowing Two Sides and the Angle in Between Area = ac ● sinB 2 Area = ab ● sinC Area = bc ● sinA A c b B C a

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 Area = ac●sinB 2 Area = 6 ●12sin(36.3°) Area = 21.3cm² 6cm 36.3° 12cm

Hero’s Formula Where ‘p’ = HALF of the perimeter Example: 2 2 8cm 6cm 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

- Find the area of the following triangle Example: - Find the area of the following triangle Area = ac●sinB 2 Area = 6 ●12sin(36.3°) Area = 21.3cm² 6cm 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 Step 4: Find mDC Step 5: Add mBD and mDC Step 6: Use any formula you wish to find the total area A 10cm 55 25 B C D

Class work and homework Math 3000 pg 233 act 1 and 2, and pg 235 numbers 1-5, 7 and hand outs