Daily Check

For the triangle at the right, find the sine, cosine, and tangent ofangle z.

X

Y Z

817

15

Math II

UNIT QUESTION: What patterns can I find in right triangles?Standard: MM2G1, MM2G2

Today’s Question:How do you use trig ratios to find the all the missing parts of a triangle?Standard: MM2G2.a,b

oppositehypotenuse

SinOpp

Hyp

Leg

adjacent

CosAdj

Hyp

Leg

TanOpp

Adj

Leg

Leg

hypotenuseopposite

adjacent

If you know the value of a specific trig ratio for an unknown angle, you can calculate the measure of the angle.

These are used to find the angle when you already know the value for the ratio. On the calculator there will be a button, sometimes it reads “2nd”, that will need to be pushed before you push the Sin, Cos or Tan button. This button will allow you to do the function above the button.

Using a Calculator

For example, if for the triangle below

B

3

8.375

SinB

SinB

8

3

On most calculators written above the Sin, Cos and Tan buttons are:

1 1 1, ,Sin Cos Tan

Using a Calculator3

8.375

SinB

SinB

10.375 22.02Sin

1 0.375Sin

On the calculator enter 0.375 the hit the “2nd” button and then the “Sin” button.

On a graphics calculators you will enter it just like it reads in the equation.

B

83

The number 22.02 should be displayed. This is the angle that has a Sin value of 0.375

Then you can calculate the angle value.

22.02B Then, that means

Problem-Solving Strategies

2 2 2 2 2 215 20 12 13

25 25

c k

c k

Scenario 1) You are given 2 sides of the triangle. Find the other side and the two acute angles.

1A. Use the Pythagorean theorem to find the 3rd side.

1B. Use an inverse trig function to get an angle. Then use that angle to calculate the 3rd angle. Sum of the angles = 180º

1 1

15 12

20 1315 12

20 13

36.9 22.6

53.1 67.4

TanA CosK

A Tan K Cos

A K

B G

OR

AC

B

20

c15

KC

G13

k

12

2A. Use an inverse trig function to get an angle. Then use the sum of the angles = 180º to find the 3rd angle.

Problem-Solving StrategiesScenario 1) You are given 2 sides of the triangle. Find the other side and the two acute angles.

1 1

15 12

20 1315 12

20 13

36.9 22.6

53.1 67.4

TanA CosK

A Tan K Cos

A K

B G

2B. Use a trig ratio using one of the two angles to get the 3rd side.

2036.9 22.6

2020

20 22.636.9

25 5

aCos Tan

c

c a TanCos

c a

OR

AC

B

20

c15

KC

G13

k

12

Problem-Solving Strategies

Scenario 2) You are given an angle and a side.

Find the other angle and the two other sides.

1A. Use 2 different trig ratios from the given angle to get each of the other two sides.

AC

B

b

26a

51

1B. Use the sum of the angles to get the 3rd angle.

51 5126 26

0.6293 26 0.7771 26

16.4 20.2

a bCos Sin

a b

a b

180 (90 51 )

39

A

A

Problem-Solving Strategies

Scenario 2) You are given an angle and a side.

Find the other angle and the two other sides. A

C

B

b

26a

51

2B. Use 2 different trig ratios from the 3rd angle to get each of the other two sides.

2A. Use the sum of the angles to get the 3rd angle.

39 3926 26

0.6293 26 0.7771 26

16.4 20.2

b aSin Cos

b a

b a

180 (90 51 )

39

A

A

Problem-Solving StrategiesScenario 3) You are given all 3 sides of the triangle.

Find the two non-right angles.

1. Use 2 different trig ratios to get each of the angles.

AC

B

24

257

1 1

24 24

25 724 24

25 7

16.3 73.7

CosA TanB

A Cos B Tan

A B

Problem-Solving StrategiesScenario 3) You are given all 3 sides of the triangle.

Find the two non-right angles.

2A. Use a trig ratio to get one angle.

AC

B

24

257

2B. Use the sum of angles to get the 3rd angle

1

24

2524

25

16.3

CosA

A Cos

A

180 (90 16.3 )

73.7

B

B

Angle of Elevation/Depression

Balloon

You

Angle of depression

Angle of elevation

Sometimes when we use right triangles to model real-life situations, we use the terms angle of elevation and angle of depression.

If you are standing on the ground and looking up at a hot air balloon, the angle that you look up from ground level is called the angle of elevation.

If someone is in the hot air balloon and looks down to the ground to see you, the angle that they have to lower their eyes, from looking straight ahead, is called the angle of depression.

Angle of Elevation/Depression

If you look up 15º to see the balloon, then the person in the balloon has to look down 15º to see you on the ground.

Notice that in this situation, the one of the legs that forms the right angle is also the height of the balloon.

Angle of elevation = Angle of depression.

Balloon

You

Angle of depression = 15º

Angle of elevation= 15º

Draw a Picture

When solving math problems, it can be very helpful to draw a picture of the situation if none is given.

Here is an example.

Find the missing sides and angles for Triangle FRY. Given that angle Y is the right angle, f = 68, and y = 88.

RY

F

68

88r

The picture helps to visualize what we know and what we want to find!