Geometry 9.6 Solving Right Triangles. July 2, 2015Geometry 9.6 Solving Right Triangles2 Goals Use inverse trig functions to find angle measures. Solve

Geometry

9.6 Solving Right Triangles

April 19, 2023 Geometry 9.6 Solving Right Triangles 2

Goals

Use inverse trig functions to find angle measures.

Solve right triangles. Solve problems using right triangles.

Inverse functions in trig

holt homework help 8.2



Solving a triangle means…

Finding the lengths of the three sides. Finding the measure of the three

angles.A

B C

b

a

c

In a right triangle, one angle is always 90 and we don’t need to worry about it.


We can use…

Trig equations Pythagorean Theorem Inverse trig functions


Inverse Trig Functions

If sin A = x, then sin-1x = A. If cos A = x, then cos-1x = A. If tan A = x, then tan-1x = A.


Example 1

Sin A = 0.7660. What is A? Sin-1(.766) = A Use 2ndsin (.766) in your calculator ***MAKE SURE YOU ARE IN DEGREE MODE********

A 50


Example 2

Cos A = 0.2079. What is A? Cos-1(.2079) = A A 78


Example 3

Tan A = 0.1051. What is A? Tan-1(.1051) = A A 6


Solving a triangle

12

7

A

B

c

First, we will find A.

tan A = 7/12

tan-1 (7/12) = A

A 30


Solving a triangle

12

7

A

B

c

Now find B.

Since A and B are complementary, B is about 60.

30


Solving a triangle

12

7

A

B

c

Find side c.

Pythagorean Theorem

30

60

2 2 2

2

2

12 7

144 49

193

13.9

c

c

c

c


Solving a triangle

12

7

A

B

13.9

The triangle is solved.

Notice: the measures are all approximate.30

60


You try it. Solve the triangle.

15

32

A

B

c

First, find angle A. tan A = 32/15

tan-1(32/15) = A

A 65



15

32

A

B

c

Next, find angle B. 90 – 65 = 25

65



15

32

A

B

c

Now find side c.

65

25

2 2 2

2

2

15 32

225 1024

1249

1249

35.3

c

c

c

c

c



15

32

A

B

35.3


65

25


Example Solve the triangle.

16.5

38

A

b

a

Find A first, since it’s the complement of the other acute angle.

A = 90 – 38 = 52

52



16.5

38

A

b

a

Now use sine to find a.

52



16.5

38

A

b

13.0

Now use cosine to find b.

52



16.5

38

A

10.2

13.0


52

22

Important You can solve a triangle in any order you

want to, as long you have the data you need for each step.

It’s best not to use rounded data in any calculation.

Be very careful using a calculator. Be sure you are in DEGREE Mode when

using your calculator! Check everything twice.


Solve this triangle.

25

10

A

c

B


Solution

25

10

A

c

B

c2 = 252 + 102

c2 = 725

c 26.9

tan B = 25/10

B = tan-1(2.5)

B ≈ 68

A = 90 – 68 = 22

26.9

68

22


Indirect Measure

One of the most powerful uses of trig is to measure things that can’t be measured directly. This is indirect measure.

Fundamental process used in surveying, map making, astronomy and other applications.


Problem Using a transit.

Jim110 ft.

7

Jim the Surveyor uses a transit to measure distances. He knows the distance between the tree and the fire hydrant is 110 ft. And to move from one to the other he swings his transit through 7. How far is he from each object?


Problem Solution

Jim110 ft.

7x

110tan 7

110 110

tan 7 .1228896

x

x

x


Problem Solution

Jim110 ft.

7896

110sin 7

110 110

sin 7 .1219902

y

y

y

y


Is this correct?

Jim110 ft.

7896

902

YES!


Indirect Measure

Jim110 ft.

7896

902

Using trig, Jim can determine the distances to the tree and the fire hydrant without measuring them directly.


Summary

Solving a triangle means to find all six parts: 3 angles, 3 sides.

Use inverse trig function (sin-1, cos-1, tan-1) to find angles.

Use given values when possible.

Documents

Geometry 9.6 Solving Right Triangles. July 2, 2015Geometry 9.6 Solving Right Triangles2 Goals Use inverse trig functions to find angle measures. Solve