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
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May 14, 2014
Apr 257:32 AM
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