Using your Calculator and Solving Trig Problems Precal D – Section 7.3 (part 2)

Using your Calculator and Solving Trig ProblemsPrecal D – Section 7.3 (part 2)

Approximate value of trig functions using your calc Make sure that your calculator is in the

correct mode (degree or radian) Find the following:

sin 27 tan(/9)

cos 0.24

Approximate value of trig functions using your calc Make sure that your calculator is in the

correct mode (degree or radian) Find the following:

sin 27 = 0.4540 tan(/9) = 0.3640

cos 0.24 = 0.9713

Approximate value of trig functions using your calc To find the value of csc, sec, and cot you

will need to remember your reciprocal identities:

tan

1cot

cos

1sec

sin

1csc

Approximate value of trig functions using your calcFor example, csc (/7)

7sin1

Enter this into your calculator (make sure the mode is correct!)

=2.3047

Approximate value of trig functions using your calcFind the value of the following:

1. csc 73

2. cot 2.35

3. sec (/13)

Approximate value of trig functions using your calcFind the value of the following:

1. csc 73 = 1.0457

2. cot 2.35 = -0.9877

3. sec (/13) = 1.2099

Angle of Elevation and Depression

The angle of elevation is measured from the horizontal up to the object.

Imagine you are standing here.

Angle of Elevation and DepressionThe angle of depression is measured from the horizontal down to the object.

Constructing a right triangle, we are able to use trig to solve the triangle.

A second similar triangle may also be formed.

Angle of Elevation and Depression

Example #1

Angle of Elevation and DepressionSuppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat?

Construct a triangle and label the known parts. Use a variable for the unknown value.

5.7o

5.7o

150 ft.

x

Angle of Elevation and DepressionSuppose the angle of depression from a lighthouse to a sailboat is 5.7o. If the lighthouse is 150 ft tall, how far away is the sailboat?

5.7o

5.7o

150 ft.

x

Set up an equation and solve.

Angle of Elevation and Depression

5.7o

150 ft.

x

150tan(5.7 )o

x

tan(5.7 ) 150ox

150

tan(5.7 )ox

Remember to use degree mode!

x is approximately 1,503 ft.

Angle of Elevation and Depression

Example #2

Angle of Elevation and DepressionA spire sits on top of the top floor of a building. From a point 500 ft. from the base of a building, the angle of elevation to the top floor of the building is 35o. The angle of elevation to the top of the spire is 38o. How tall is the spire?

Construct the required triangles and label.

500 ft.

38o 35o

Angle of Elevation and DepressionWrite an equation and solve.

Total height (t) = building height (b) + spire height (s)

500 ft.

38o 35o

Solve for the spire height.

t

b

s

Total Height

tan(38 )500

o t

500 tan(38 )o t

Angle of Elevation and DepressionWrite an equation and solve.

500 ft.

38o 35o

Building Height

tan(35 )500

o b

500 tan(35 )o b t

b

s

Angle of Elevation and Depression

5050 0 t0 tan an(3(38 ) 5 )o o s

Write an equation and solve.

500 ft.

38o 35o

500 tan(38 )o t 500 tan(35 )o b

5050 0 t0 tan an(3(38 ) 5 )o o s

The height of the spire is approximately 41 feet.

t

b

s

Total height (t) = building height (b) + spire height (s)

Angle of Elevation and Depression

Example #3

Angle of Elevation and Depression

A hiker measures the angle of elevation to a mountain peak in the distance at 28o. Moving 1,500 ft closer on a level surface, the angle of elevation is measured to be 29o. How much higher is the mountain peak than the hiker?

Construct a diagram and label.

1st measurement 28o.

2nd measurement 1,500 ft closer is 29o.

Angle of Elevation and Depression

Adding labels to the diagram, we need to find h.

28o 29o

1500 ft x ft

h ft

Write an equation for each triangle. Remember, we can only solve right triangles. The base of the triangle with an angle of 28o is 1500 + x.

tan 281500

o h

x

tan 29o

h

x

Angle of Elevation and Depression

tan 29oh

x

tan(29 )ox h

Now we have two equations with two variables.Solve by substitution.

tan 281500

o h

x

(1500 ) tan(28 )ox h

(1500 ) tan(28 ) tan(29 )o ox x

Solve each equation for h.

Substitute.

Angle of Elevation and Depression

1500 tan(28 ) tan(28 ) tan(29 )o o ox x

(1500 ) tan(28 ) tan(29 )o ox x Solve for x. Distribute.

Get the x’s on one side and factor out the x.

Divide.

1500 tan(28 ) tan(29 ) tan(28 )o o ox x

1500 tan(28 ) tan(29 ) tan(28 )o o ox

1500 tan(28 )

tan(29 ) tan(28 )

o

o ox

x = 35,291 ft.

Angle of Elevation and Depression

tan(29 )ox h

However, we were to find the height of the mountain. Use one of the equations solved for “h” to solve for the height.

x = 35,291 ft.

(35,291) tan(29 ) 19,562o

The height of the mountain above the hiker is 19,562 ft.

Finding Value of Trig FunctionsAngle of Elevation and Depression

Assignment 7.3 p. 536 #11, 12, 14, 16, 17, 21, 23, 25, 28, 30, 33, 35, 37, 38, 40, 42, 44, 57, 60, 63, 66, 73, 76

Remember to change your calculator between radians and degrees when required.