296

1 2

y

2y912

2y21 )21,14( J

The midpoint of is M(-4,6). If point R is(6, -9), find point J.

RJ

2

y9,

2

x6)6,4( 22

264

1 2

x

2x68

2x14

1 2 1 2x x y yM ,

2 2

x

y

Determine whether the points (3, 4), (5, 1), and(2, -1) are vertices of a right triangle.

(3,4)

(5,1)

(2,-1)

Yes

2 2

1D 3 2 4 1

2 2

1D 1 5

1D 1 25 26

2 2

2D 5 3 1 4

2 2

2D 2 3

2D 4 9 13

2 2

3D 5 2 1 1

2 2

3D 3 2

3D 9 4 132 2 2a b c

2 2 213 13 26

13 13 26

1D 2D

3D

Trigonometric Functions with Right Triangles

Trig functions will be done involving a right triangle.Trig functions are always done using one of the non-right anglesWe set up ratios using the 3 sides of the triangle and solve for missing parts

A

B C

Trigonometric Functions with Right Triangles

hypotenuse

The sides have names that we need to know to set up the ratiosThe hypotenuse is always across from the right angle (we should already know this)

A

B C

Trigonometric Functions with Right Triangles

hypotenuse

The other 2 sides names depend on which angle we are working withFor this problem we will say we are working with angle C

A

B C

Once we have established this – we can label the other 2 sides

Trigonometric Functions with Right Triangles

hypotenuse

The opposite side is across (or opposite) the angle that we are working withThe remaining side is called the adjacent side

A

B C

opposite

adjacent

Trigonometric Functions with Right Triangles

hypotenuse

There are 3 trig functions we will work with – sine (sin), cosine (cos), and tangent (tan)

A

B C

opposite

adjacent

sinopposite

Chypotenuse

cos

adjacentC

hypotenuse

tanopposite

Cadjacent

Setting up the trig functions with numbers

5

The sides still have the same names – we just need to plug the numbers in

A

B C

3

4

3sin

5C

4cos

5C

3tan

4C

hypotenuseopposite

adjacent

Find the sin C, cos C, tan C

Setting up the trig functions with numbers

5

The only difference is the angle we are working with is now A – the sides change

A

B C

3

4

hypotenuse

opposite

adjacent

4sin

5A

3cos

5A

4tan

3A

Find the sin A, cos A, tan A

Trigonometric Functions with Right Triangles

The calculator has a sin, cos, and tan button on it. Make sure your calculator is in degree mode before using it

A

B C

You can use this button to find the value of an angle to help you solve for missing sides

Trigonometric Functions with Right Triangles

Set up the ratio using the information you have (4 is the opposite side and x is the adjacent – so use tangent)

A

B C

Find side AB in the following triangle (round to the nearest tenth)

25º

9

x

9tan 25

x

In your calculator get the value for tan 25 (4 decimals)

tan 25 9

1 x

Cross multiply and solve

.4663 9

1 x

.4663 9x

.4663 .4663

19.3x

Trigonometric Functions with Right TrianglesA

B C

Find the missing sides60º

9

x9

sin 60y

y

opp

hypadj

cos60x

y

9tan 60

x

sin 60 9

1 y

sin 60* 9y .8660* 9y .8660 .8660

10.39y

tan 60 9

1 x

tan 60* 9x

1.7321 1.7231

1.7321* 9x

5.20x

Trigonometric Functions with Right Triangles

A

B C

35º

210

x

opphyp

adj

You are a crane operator with a 210 foot boom. You lift up a rock so that it is directly in front of you. The boom is at an angle of 35º. How far in front of you is the rock?

cos35

1 210

x

.8192*210 x172.032 x