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Right Triangle Trigonometry. By: Jeffrey Bivin Lake Zurich High School jeff.bivin@lz95.org. Last Updated: December 1, 2010. SOH CAH TOA. hypotenuse. opposite. θ. adjacent. Reciprocal Identities. hypotenuse. opposite. θ. adjacent. Find the sides. B. 2. c. 1. a. A. C. b.

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right triangle trigonometry

Right Triangle Trigonometry

By: Jeffrey Bivin

Lake Zurich High School

jeff.bivin@lz95.org

Last Updated: December 1, 2010

soh cah toa
SOH CAH TOA

hypotenuse

opposite

θ

adjacent

reciprocal identities
Reciprocal Identities

hypotenuse

opposite

θ

adjacent

find the sides4
Find the sides.

Use your

calculator!

B

15

c

68o

a

22o

A

C

b

find the sides5
Find the sides.

B

c

56o

25

a

34o

A

C

b

quotient identities
Quotient Identities

hypotenuse

opposite

We Know

θ

adjacent

pythagorean identities
Pythagorean Identities

hypotenuse

opposite

θ

adjacent

Divide by hyp2

Divide by adj2

Divide by opp2

using pythagorean identities
Using Pythagorean Identities
  • Find cosθ if sinθ =
using pythagorean identities1
Using Pythagorean Identities
  • Find secθ if tanθ =
using pythagorean identities2
Using Pythagorean Identities
  • Find sinθ if cotθ =
co function identities
Co-function Identities

Use your calculators to evaluate each of the following.

cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)

Complimentary Angles

co function identities1
Co-function Identities

cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)

sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ)

Complimentary Angles

co function identities2
Co-function Identities

cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)

sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ)

tan(θ) = cot(90o – θ) and cot(θ) = tan(90o – θ)

Complimentary Angles

co function identities3
Co-function Identities

cos(θ) = sin(90o – θ) and sin(θ) = cos(90o – θ)

sec(θ) = csc(90o – θ) and csc(θ) = sec(90o – θ)

tan(θ) = cot(90o – θ) and cot(θ) = tan(90o – θ)

Complimentary Angles

90o- θ

c

a

θ

b

slide22

t

(b, a)

(a, b)

90o-t

t

-t

(a, -b)

slide24

A surveyor is standing 45 feet from the base of a large tree. The surveyor measures the angle of elevation from the ground to the top of the tree to be 67.5o. Find the height of the tree.

h

67.5o

45 feet

slide25

An airplane flying at 4500 feet is on a flight path directly toward an observer. If 30o is the angle of elevation from the observer to the plane, find the distance from the observer to the plane.

d

4500 feet

30o

slide26

In traveling across flat land a driver noticed a mountain directly in front of the car. The angle of elevation to the peek is 4o. After the driver traveled 10 miles, the angle of elevation was 11o. Approximate the height of the mountain.

h

11o

4o

x

10 mi

10 + x

slide27

A flagpole at the top of a tall building (and at the edge of the building) is know to be 45 feet tall. If a man standing down the street from the building calculates the angle of elevation to the top of the building to be 55o and the angle of elevation to the top of the flagpole to be 57o. Find the height of the building.

45

h

57o

55o

d

slide28

An observer standing on the cliff adjacent to the ocean looks out and sees an airplane flying directly over a ship. The observer calculates the angle of elevation to the plane to be 14o and the angle of depression to the ship to be 27o. How high above the ship is the airplane if we know that the ship is 1.5 miles from shore?

p

14o

1.5 mi

27o

b

Distance of plane above ship = p +b=

slide29

In Washington, D.C., the Washington Monument is situated between the Capitol and the Lincoln Memorial. A tourist standing at the Lincoln Memorial tilts her head at an angle of 7.491° in order to look up to the top of the Washington Monument. At the same time, another tourist standing at the Capitol steps tilts his head at a 5.463° to also look at the top of the Washington Monument. Find the distance from the Lincoln Memorial to the Washington Monument.