5.9.3 Problem Solving with the Pythagorean Theorem and

Trigonometry

5.9.4 Proving the Pythagorean Identity

• Angle of elevation- the angle created by a horizontal line and an upward line of sight to an object above the observer.

• Angle of Depression- the angle created by a horizontal line and a downward line of sight to an object below the observer.

Angles of Elevation & Depression

Example 1

The height of a tree is 15 meters. To the nearest whole degree,

what is the angle of elevation of the sun when the tree casts a

shadow that is 9.3 meters long on level ground?

Example 2

A meteorologist reads radio signals to get information from a

weather balloon. The last alert indicated that the angle of

depression of the weather balloon to the meteorologist was

41º and the balloon was 1,810 meters away from his location on

the diagonal. To the nearest meter, how high above the ground

was the balloon?

Example 3

A sonar operator on an anchored cruiser detects a pod of

dolphins feeding at a depth of about 255 meters directly below.

If the cruiser travels 450 meters west and the dolphins remain at

The same depth to feed, what is the angle of depression, x, from

the cruiser to the pod? What is the distance, y, between the

cruiser and the pod? Round your answers to the nearest whole

number.

5.9.4- Example 1

Prove the Pythagorean identity sin2θ + cos2θ =1

Vocabulary

• Pythagorean identity: sin2θ + cos2θ = 1

• Note that sin2θ is the same as (sinθ )2.

• For example, the identity could be written sin2θ = 1 – cos2θ or cos2θ = 1 – sin2θ.

• Two more identities called ratio identities help with simplifying:

tanq =

sinq

cosq and cotq =

cosq

sinq

Vocabulary

• You can solve this Pythagorean identity for one of the trigonometric functions. For example:

sin2 q + cos2 q = 1

sin2 q = 1- cos2 q

sinq = 1- cos2 q

Example 2 If sinθ= 2/5 and 0 < θ < 90, use the Pythagorean identity

sin2θ + cos2θ = 1 to find the values of cosθ and tanθ.

Extra Practice 1. Simplify √56

2. Simplify (√3)/ (√2)

3. If sinθ= 2/9 and 0 < θ < 90, use the Pythagorean identity sin2θ + cos2θ = 1 to find the value of cosθ.

Example 3

Simplify the expression sin2θ – sin2θcos2θ.

Example 4 Mackenzie is flying a kite. She is standing on the handle of the

kite string for a minute while she ties her shoelace. The angle

that the kite string makes with the ground is θ. If the length of

The kite string is 230 feet when the kite is 190 feet off the

ground, find the cosine of θ.