1.51.5Solving Right TrianglesSolving Right Triangles

JMerrill, 2005JMerrill, 2005Revised 2009Revised 2009

Recall from Last WeekRecall from Last Week

SOH-CAH-TOASOH-CAH-TOA• Sine:Sine: abbreviation: abbreviation: ssin =in = • Cosine:Cosine: abbreviation: abbreviation: ccos = os = • Tangent:Tangent: abbreviation: abbreviation: ttan =an =

• These are the ratios of 2 sides. These are the ratios of 2 sides.

ppositeypoteO

H nuse

djacentypoteA

H nuse

OA

ppositedjacent

Evaluating with a Calculator Evaluating with a Calculator When When

• Mode is in degrees (for now!)Mode is in degrees (for now!)• Solve: Solve:

– cos 41cos 41o o = = tan 87tan 87oo = =

– sin 18sin 18oo = = cos 180cos 180o o ==

• These are still the ratios of 2 sides—in These are still the ratios of 2 sides—in decimal form.decimal form.

0.7550.309

19.081 1

Application—Finding the Application—Finding the Length of a SideLength of a Side• A plane is one mile A plane is one mile

above sea level above sea level when it begins to when it begins to climb at a constant climb at a constant angle of 2angle of 2 for the for the next 70 ground next 70 ground miles. How far miles. How far above sea level is above sea level is the plane after its the plane after its climb?climb?

• So, about 3 ½ miles total.So, about 3 ½ miles total.

2o

70

x

tan2 70(70)tan2 (70)702.444

o

o

x

x

x

You do:You do:• A 16-foot ladder is A 16-foot ladder is

propped against a propped against a building. The angle building. The angle it forms with the it forms with the ground measures ground measures 5555. How far up the . How far up the side of the building side of the building does the ladder does the ladder reach?reach?

55o

16x

sin55 1616sin5513.106

, 13.1

.

x

xx

So the ladder reachesabout f eet up thebuilding

Finding an AngleFinding an AngleTo find the measure of an To find the measure of an

angleangle you must take the you must take the inverse of the function. inverse of the function. The inverse tangent, for The inverse tangent, for example, is written tanexample, is written tan-1-1. . On your calculator, you On your calculator, you push 2push 2ndnd tan. tan. Anytime you Anytime you are looking for the angle, are looking for the angle, you push the 2you push the 2ndnd button! button!

θθ is the Greek symbol for is the Greek symbol for theta (lower case). It will theta (lower case). It will be used to represent an be used to represent an angle in degrees.angle in degrees.

553553

100100

θθ

1

553tan 100553tan 79.7100

o

Angle of ElevationAngle of Elevation• Angle of elevationAngle of elevation is the angle is the angle

formed by the horizontal line of sight formed by the horizontal line of sight to an object that is above the to an object that is above the horizontal line.horizontal line.

Angle of DepressionAngle of Depression• Angle of depressionAngle of depression is the angle is the angle

formed by the horizontal line of sight formed by the horizontal line of sight to an object that is below the to an object that is below the horizontal line.horizontal line.

You do:You do:

SolutionsSolutions• How far are you How far are you

from the ground?from the ground?• How tall is the How tall is the

pole?pole?

tan15 7575tan15 2

0.1

o y

yy

tan25 7575tan25 34.97

o x

xx

You are approximately 20.1 feet from the ground. From your position to the top of the pole is approximately 34.97 feet. Therefore, 20.1 + 34.97 = 55.07. Thus, the pole is about 55.07 feet tall.

NavigationNavigation• Direction is often given as a measure Direction is often given as a measure

of an acute angle with respect to the of an acute angle with respect to the north-south vertical line. A bearing (or north-south vertical line. A bearing (or heading) of N 20heading) of N 20oo E means that the E means that the plane/ship is bearing 20plane/ship is bearing 20oo to the east of to the east of north.north.

• So, S 48So, S 48o o W means? W means? • 4848o o to the west of southto the west of south

ExampleExample• A jet takes off on a bearing of N 28A jet takes off on a bearing of N 28oo

E, flies 5 miles, then makes a left E, flies 5 miles, then makes a left (90(90oo) turn and flies 12 miles. If the ) turn and flies 12 miles. If the control tower operator wanted to control tower operator wanted to locate the plane, what bearing would locate the plane, what bearing would she use? she use?