Precalc - Chapter 4 Notes PDF

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5.5 The Law of Sines

Pre-Calculus

Learning Targets: 1. Use the Law of Sines to solve non-right triangles. Today we will focus on solving for the sides and angles of non-right triangles when given two angles and a side.

Derivation: The Law of Sines

Example 1: In ∆ABC , A  49 , a = 32 and B  20. Solve ∆ABC.

Example 2: The bearings of two ramps on the shore from a boat are 115 and 123. Assume the two ramps are 855 feet apart. How far is the boat from the nearest ramp on shore if the shore is straight and runs north-south?

5–1

5.6 The Law of Cosines

Pre-Calculus

Learning Targets: 1. Use the Law of Cosines to solve non-right triangles. 2. Find the area of a non-right triangle when given SAS. 3. Find the area of a non-right triangle when given SSS using Heron’s Formula. The Law of Sines works well when we are given AAS, ASA or when we have a “magic pair”. But what if we are not able to find an angle and the side across from it? What if we are given _________ or _________? The Law of Cosines: a  b  c  2 bc cos A  Note that we must know two sides to use this law.  We must also know either the angle A between the two sides (SAS) or the third side a.  Once a problem is started with Law of Cosines, you SHOULD continue with this law to find all missing pieces! 2

2

2

Example 1: Solve ∆WXY if x = 17cm, y = 6.2 cm and W = 50°.

Area of a Non-Right Triangle: Area  21 ab sin C Example 2: Find the area of ∆ABC when A  49 , c = 13 and b = 18.

While Area  12 ab sin C works to find the area when given SAS what if we are given SSS? Heron’s Formula: When given a, b, and c in a non-right triangle… Semiperemeter : S 

1 2

a  b  c 

Area 

s  s  a  s  b s  c

Example 3: Bob wants to sod a portion of his backyard roughly in the shape of a triangle with sides 9 feet, 12 feet and 15 feet. How many 4.5 square foot sod rolls does Bob need to buy?

5–2

5-5 The Law of Sines: Ambiguous Case

Pre Calculus

Learning Targets: 1. Determine when a triangle is not possible using Geometry and the Law of Sines. 2. Determine when a situation yields two triangles and solve for BOTH triangles using the Law of Sines. So, what about SSA? We never used SSA in Geometry! Remember AAS, ASA, SAS, and SSS were theorems in Geometry because those pieces always created two congruent Triangles. Use the given pieces of a triangle (already drawn below) and a ruler to try and create triangles. Notice one side is dashed because its length is unknown.

B

AB=4.5cm A  30  BC=3cm A

The problem with Two Sides and a NON-Included Angle is there is more than one possibility for your answers! We call it the “ambiguous case”. No ∆’s:

1 ∆:

2 ∆’s:

Example 1: Determine from the given information if zero, one or two triangles may exist. Explain! a) W = 56°, w = 30, x = 26

b) R = 125°, r = 6, s = 14

Example 2: Solve the triangle in Example 3c.





5–3

c) A = 38°, b = 21, a = 14

Fundamental Identities – Day 1

Pre-Calculus

Learning Targets: 1. Know the fundamental identities: reciprocal, quotient, Pythagorean, cofunction and even-odd. 2. Rewrite trigonometric expressions using the following techniques: a. Rewrite with sine and/or cosine b. Use the fundamental identities listed in target #1 c. Factor with the GCF 3. PROVE trigonometric identities using all of the above. For more background information, read the Chapter 5 Overview (pgs 443-444) …  This chapter is particularly important for those continuing on into college mathematics. 

We are now shifting our focus from problem solving to theory and proof.



We are studying the connections among the trigonometric functions themselves.

Identities: Mathematical sentences that are true for all values of the variable for which both sides of the equation are defined. Truth statements. Similar to postulates and theorems in Geometry.

 Reciprocal Identities:

1 sin  1 sin   csc

csc  

1 cos  1 cos  sec 

1 tan  1 tan   cot 

sec  

cot  

 Quotient Identities:

tan  

sin cos 

cot  

cos sin 

We already know these Identities, but are there others? Since we began our understanding of Trigonometry with the Unit Circle and right triangles, let’s return there and look for patterns. 1. Can we relate x & y?

2. How are the two acute angles related?

5–4

 Pythagorean Identities:

cos2   sin 2   1

 Cofunction Identities (aka Complementary Angles):

sin  2    

cos  2    

tan  2    

cot  2    

sec  2    

csc  2    

 Odd-Even Identities:

sin    

cos    

tan    

csc    

sec    

cot    

To simplify or rewrite trigonometric expressions:  The final answer should be simpler than the original expression.  Prefer one trig. function.  Prefer NO fractions.  If you are stuck try the following strategies (more strategies in the next lesson) I. By changing to sines and cosines. . . 1. cot  sec  sin 

II. By using Pythagorean and/or Co-Function identities. . .

    2   1 cos  2 

2. sin   cos  2

5–5

III. By factoring the GCF. . . 2 3. cos  cos sin 

IV. By using Even-Odd and Reciprocal identities 4. sin( x)csc(x )

Proving Identities From Geometry, a proof is a series of statements, facts, definitions, postulates, etc organized in a logical order to deductively reason that a conclusion follows from a given statement(s). In Trigonometric proofs, we do the same thing using identities but we organize our proof as a series of algebraic manipulations that are “sufficiently obvious enough to require no additional justifications” according to your book. In other words, start with the most complex side of the equation and use identities to write equivalent statements until you get to the other side of the original equation. You will use some of the same strategies we used in simplifying.

Prove each identity. 2 5. cot  

csc  cot  sec

6.

5–6

sec   x  1  tan x 2

 cos x

Fundamental Identities – Day 2

Pre-Calculus

Learning Targets: 1.

2.

Rewrite trigonometric expressions using the following additional techniques: d. Combine with Least Common Denominator (LCD) e. Factor with Difference of Squares f. Factor using trinomials (the box) PROVE trigonometric identities using all of the above.

V. By combining with LCD. . . 1 1 7.  1  cos x 1  cos x

8. sec x  sin x sin x

cosx

VI. By Factoring with the Difference of Squares. . . 9.

1  cos2  1 cos

10.

sec x  1 tan 2 x

VII. By Factoring with trinomials with a box. . . 4 2 11. cos x sin x  2 cos x sin x  cos x

Remember you can use any of the above strategies and/or any of the strategies from day 1 to PROVE! Prove each identity. cos 1  sin 12.   2sec  1  sin cos

5–7

5.3 Sum and Difference Identities

Pre-Calculus

Learning Targets: 1. 2. 3.

Use the sum and difference identities to prove other identities. Use the sum and difference identities to rewrite an expression as the sine, cosine and/or tangent of a single angle. Use the sum and difference identities to find the exact value of an expression.

 Sum and Difference Formulas

sin  a  b   sin a cos b  cos a sin b Remember:

cos  a  b   cos a cos b  sin a sin b

1. Sine Sign ___________________.

sin  a  b  

2. Cosine Sign __________________.

cos  a  b   tan  a  b  

tan a tan b

3. Tangent

1  tan a tan b

.

tan  a  b  

Example 1: Prove each of the following is an identity.

     cos  2 

a) cos       cos 

b) sin 

Example 2: Write the expression as the sine, cosine or tangent of a single angle. Then, evaluate if possible.

a) sin 95 cos 55 + cos 95 sin 55

b) cos 



       cos    sin   sin   4 3  4  3

5–8

c)

tan 20   tan 34  1  tan 20  tan 34 

Example 3: Use the sum and difference identities to find the exact value of each function. a) sin (15)

c)

 5    12 

sin 

b) cos (165)

d) tan  15 

5–9

5.4 Multiple–Angle Identities

Pre-Calculus

Learning Targets: 1. 2. 3.

Use the double angle identities to prove other identities. Use the double angle identities to rewrite an expression as the sine, cosine and/or tangent of a single angle. Use the double angle identities to find the exact value of an expression.

 While there are several identities in this section, we will only concern ourselves with the Double Angle identities which come straight from the Angle Sum Identities. Derivation 1: Using sin(   )  sin  cos   cos  sin  …

Derivation 2: Using cos(   )  cos  cos   sin  sin  …

2 2 Or using the Pythagorean Identity: cos x  sin x  1

Derivation 3: Using tan(   ) 

tan   tan  1 tan  tan 

…

Example 1: Verify the following identities: b) sin 2   2 tan  1 tan2 

a) 1  cos 2  cot  sin 2

5–10

Example 2: Write as the function of one angle. Simplify, if you can, without using a calculator. a) 2 sin15 cos15

b) 2 cos2   1 8

2 c) 1  2sin 120

Example 3: If sin A  a) cos 2A

d)

12 13

2tan 15 1  tan 2 15

, and angle A is in the first quadrant, determine: b) tan 2A

    5–11

Solving Trigonometric Equations

Pre-Calculus

Learning Targets: 1. Solve trigonometric equations using identities. 2. Find the correct solution(s) using the given domain and/or inverse notation.  To Solve Trig Equations: 1. Use identities to get one trigonometric function, if possible. 2. Identify your variable. 

2 If x only OR x only … isolate x.

 If x and x 2 … factor so you can use the Zero Product Property to isolate x. Use the inverse to find the missing angle. Give the correct solution(s) based on the directions and context of the problems.

3. 4.

Example 1: Solve each equation for [0, 2π). a) tan x  tan x  0 2

b) 2sin x 5sin x 2  0

c) sin 2x  2 cos x

b) 2cos2 x  cosx  cos 2x

c) 4sin2 x  4sin x 1  0

2

Example 2: Find all real solutions. 2 a) 2sin x 1  0

5–12...