Problem Solving with Right Triangles

Section 2

Lesson Objectives:

You will be able to:

1. Find missing angles and sides using trigonometric ratios

2. Use the angle of elevation and angle of depression to solve real world problems

Lesson Objective # 1:

Find missing angles and sides using trigonometric ratios

Solving Right TrianglesSolving Right Triangles

• With these six ratios, it is possible – to solve for any unknown side of the right triangle, if

another side and an acute angle are known, or – to find the angle if two sides are known.

Once upon a time, students had to rely on tables to look up these values. Now the sine, cosine, and tangent of an angle can be found on your calculator.

Once upon a time, students had to rely on tables to look up these values. Now the sine, cosine, and tangent of an angle can be found on your calculator.

Using a Calculator

AC

B

Each trig ratios has a specific value for every angle. Using a calculator we can find the decimal values for the

Sine, Cosine and Tangent ratios for any angle.

45 0.7071

81 0.1564

23 0.4245

Sin

Cos

Tan

Remember to make sure that the calculator is in Degree mode when calculating these values. This is shown as a

D, or Deg in the display.

For Example,

Inverse Trig Functions

If sin is a trig function

then sin-1 is aninverse trig function

:inverse trig functions simply “undo” trig functions

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.

Using a Calculator

For example, if for the triangle

3

8.375

SinB

SinB

B

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

Using a Calculator

1If 0.951 then 0.951CosB B Cos

1If 2.54 then 2.54TanC C Tan

1If 0.0872 then 0.0872SinJ J Sin

1If 0.4562 then 0.4562SinA A Sin

Here are some examples

1If 0.858 then 0.858CosW W Cos

1If 1.53 then 1.53TanQ Q Tan

so, 27.1A

so, 18B

so, 68.5C

so, 56.8Q

so, 5J

so, 30.9W

Finding Missing Sides

2 2 2 2 2 215 20 12 13

25 25

c k

c k

You are given 2 sides of the triangle. Find the other side and the two non-right 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.

Finding Missing SidesScenario 1) You are given 2 sides of the triangle. Find the other side and the two non-right 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

a

b

c

To solve a right triangle means to find the missing lengths of its sides and the measurements of its angles.

Problem-Solving StrategiesYou 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 StrategiesYou 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

10

15c

q

Solve for the lengthof the hypotenuseand the angle, q.

Wheel Chair Ramp• The most common question when considering a portable wheelchair ramp is: what length

of ramp is needed to achieve a safe, practical angle?  The Americans with Disabilities Act standard states that a ramp's maximum incline should be no greater than a 6 degree angle.

Assignment

• Practice Worksheet