5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 21­4:09 AM

Objective:

Manipulate trigonometric properties to verify, prove, and understand trigonmetric relationships.

Apr 21­4:10 AM

Warm-up:

Determine the exact value of the following (without a calculator):

sin π/3

cos -7π/6

tan 7π/4

sin 13π/6

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 21­4:14 AM

Notation:

sin(x) sin(x) = (sin(x))2 = sin2x

(cos(x))2 = cos2x

(tan(x))2 = tan2x

1 1 = csc(x) csc(x) = csc2x sin(x) sin(x)

Similarly.... sec2x ....... cot2x

Apr 21­4:21 AM

However:

sinx2 = sin (x2)......so be careful ...where is the 2 ?

Determine the values

sin2(π/6)

sin (30º)2

1/4

0

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 21­4:27 AM

We use identities for three basic tasks:

1. express one function in terms of another

2. simplify expressions

3. verify identities

These three tasks build our skills so that we can solve trigonometric equations later on.

Apr 21­4:27 AM

We've seen & Used the Reciprocal Identities:

We've used one of the Tangent-Cotangent Identities:

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 21­4:33 AM

Pythagorean Identities:

(cos, sin)

Apr 21­4:35 AM

For the other 2 Pythagorean Identities,

manipulate the first one:

instead of sin2θ make csc2θ

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 21­4:38 AM

You will have a quiz on the identities...

Apr 21­4:39 AM

Let's manipulate some....express sin θ in terms of cos θ

sin2θ + cos2θ = 1

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 21­4:41 AM

express tan θ in terms of sin θ1 + cot2θ = csc2θ

Apr 21­4:43 AM

Homework:

Pg 385 # 43-47 odd

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 22­4:23 PM

Objectives:Manipulate trig identities.Evaluate values using circular trig definitions.Solve application problems.

Apr 22­4:25 PM

Warm up:

Write down the Fundamental Trig Identities

Reciprocal:Cotangent-Tangent:Pythagorean:

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 22­4:26 PM

at the end of class you will have a quiz on the Fundamental Identities....use them in class...keep them handy

Homework Review:

Pg 385 # 43-47 odd

Apr 23­7:32 AM

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 23­7:37 AM

Apr 22­4:11 PM

Verify the identity by transforming the left hand side into the right hand side. 

tan2θ cosθ = sin2θ secθ

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 22­4:11 PM

Verify the identity by transforming the left hand side into the right hand side. 

secθ - cosθ = tanθ sinθ

Apr 22­4:11 PM

Verify the identity by transforming the left hand side into the right hand side. 

log csc θ = -log sin θ

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 21­7:56 AM

express sin θ in terms of cot θ 

Apr 22­4:19 PM

The point P(3, ‑4) is on the terminal ray of an angle in standard position.  What are the values of the trig  functions of this angle?

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 22­4:08 PM

The point P(‑15, 8) is on the terminal ray of an angle in standard position.  What are the values of the trig  functions of this angle?

Apr 22­4:19 PM

If sin Θ = 1/2, and the terminal side lies in Quadrant II, find cosΘ, tan Θ, cscΘ, cot Θ, and sec Θ .

If sin Θ = , find possible values for x, y, & r.

Then possible values for cosΘ, tan Θ.

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 22­4:09 PM

An angle θ is in standard position with its terminal ray in quadrant III on the line y = 3x.  What are the values of the 

trig functions of the angle?

Apr 22­4:10 PM

If cos θ > 0 and sin θ < 0, in what quadrant does the terminal ray of θ lie? 

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 22­4:18 PM

If sin θ = 3/5 and tan θ < 0, find the values of the other trig functions.  

Mar 7­8:36 AM

There are many other applications that require trig solutions. For example, surveyors use special instruments to find the measures of angles of elevation and angles of depression.

An angle of elevation is the angle between a horizontal line and the line of sight from an observer to an object at a higher level.

An angle of depression is the angle between a horizontal line and the line of sight from the observer to an object at a lower level.

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Mar 7­8:36 AM

A tower 250 meters high casts a shadow 176 meters long. Find the angle of elevation of the sun to the nearest minute.

Mar 7­8:37 AM

You just crossed a bridge 112' across from one cliff to another. Your angle of depression to the river below is 39o. How far away are you from the river?

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Mar 7­8:30 AM

Solve the right triangle ABC. Round angle measure to the nearest degree and side measures to the nearest tenth.

C

A

B

c

12

5

Mar 7­8:34 AM

You can use right triangle trig to solve problems involving geometric figures. The apothem of a regular polygon is the measure of a line segment from the center of the polygon to the midpoint of one of its sides.

Example 2: A regular hexagon is inscribed in a circle with diameter 7.52 cm. Find the apothem. Round the answer to the nearest hundredth.

7.52 cm

a

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Mar 7­8:37 AM

Example 1: The longest truck-mounted ladderused by the Dallas Fire Department is 108 feetlong and consists of four hydraulic sections.Gerald Travis, aerial expert for the dept., indicates that the optimum operating angleof this ladder is 60 . Outriggers, with an18 ft. span between each, are used to stabilizethe ladder truck and permit operating anglesgreater than 60 , allowing the ladder truckto be closer to buildings in downtown Dallas.

Assuming the ladder is mounted 8 ft off theground, how far from an 84-ft burning buildingshould the ladder be placed to achieve theoptimum operating angle of 60 ?

How far should the ladder be extended to reachthe roof?

Apr 21­4:46 AM

In class identities practice:Pg 385 # 39 (hint sum of cubes: a3 + b3 = (a + b)(a2 ­ ab + b2) , 50­56 even

Homework: Pg 385 #s 3, 4, 9 (rt triangle review)11, 13, 15 (special rt triangles, or trig values)23, 25, 26 (angle of elevation)35, 37 (pyth. id)41 (simplify)49­57 odd (verify identities)

Next class ­ review & quizin addition to the homework, know:identities, exact values (unit circle values),right triangle trig, aka:SOHCAHTOA

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 23­8:35 AM

Apr 25­7:26 AM

5.2­5.4 Identities_JB A Block.notebook

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May 14, 2014

Apr 25­7:32 AM

Apr 25­7:36 AM