Chapter 7

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Transcript Chapter 7

5-Minute Check Lesson 7-1A
Chapter 7
Trigonometric Identities and Equations
Section 7.1
Basic Trigonometric Identities
Definitions
Identity – A statement of equality between two expressions that is true for all values
of the variable(s) for which the expressions are defined.
ex: π‘₯ 2 βˆ’ 𝑦 2 = π‘₯ βˆ’ 𝑦 π‘₯ + 𝑦
Trigonometric Identity – is an identity involving trigonometric expressions.
ex:
sin πœƒ
cos πœƒ
= tan πœƒ
Reciprocal functions – (We talked about this previously… this is just review)
𝐬𝐒𝐧 𝜽 =
𝟏
𝐜𝐬𝐜 𝜽
𝐜𝐨𝐬 𝜽 =
𝟏
𝐬𝐞𝐜 𝜽
𝐬𝐞𝐜 𝜽 =
𝟏
𝐜𝐨𝐬 𝜽
𝐭𝐚𝐧 𝜽 =
𝟏
𝐜𝐨𝐭 𝜽
𝐜𝐨𝐭 𝜽 =
𝟏
𝐭𝐚𝐧 𝜽
𝐜𝐬𝐜 𝜽 =
𝟏
𝐬𝐒𝐧 𝜽
Opposite Angle Identities
𝐬𝐒𝐧(βˆ’πœ½) = βˆ’ 𝐬𝐒𝐧 𝜽
𝐜𝐨𝐬 βˆ’πœ½ = 𝐜𝐨𝐬 𝜽
Definitions continued
Quotient Identities –
𝐬𝐒𝐧 𝜽
𝐜𝐨𝐬 𝜽
= 𝐭𝐚𝐧 𝜽
𝐜𝐨𝐬 𝜽
𝐬𝐒𝐧 𝜽
Pythagorean Identities –
Recall with the unit circle we knew : π‘₯ 2 + 𝑦 2 = 1
And we know : cos πœƒ = π‘₯, sin πœƒ = 𝑦
By substitution : π‘π‘œπ‘  2 πœƒ + 𝑠𝑖𝑛2 πœƒ = 1
Which we write as:
π’”π’Šπ’πŸ 𝜽 + π’„π’π’”πŸ 𝜽 = 𝟏
If we divide both sides by sine…
𝟏 + π’„π’π’•πŸ 𝜽 = π’„π’”π’„πŸ 𝜽
If we divide both sides by cosine…
π’•π’‚π’πŸ 𝜽 + 𝟏 = π’”π’†π’„πŸ 𝜽
= 𝐜𝐨𝐭 𝜽
Examples
Use the given information to find the trigonometric values
3
2
1. If sec πœƒ = , find cos πœƒ.
4
3
2. If csc πœƒ = , find tan πœƒ when cos πœƒ > 0
1
5
3. If sin πœƒ = βˆ’ , find cos πœƒ when tan πœƒ < 0
Prove that each equation is not a trigonometric identify by producing a counterexample
1. sin πœƒ cos πœƒ = cot πœƒ
2.
sec πœƒ
tan πœƒ
= sin πœƒ
3. sin πœƒ + cos πœƒ = 1πœƒ
More Examples:
Simplify:
sin π‘₯ + sin π‘₯ π‘π‘œπ‘‘ 2 π‘₯
1. Look for any GCFs: * 𝑠𝑖𝑛π‘₯ βˆ—
sin π‘₯ (1 + π‘π‘œπ‘‘ 2 π‘₯)
2. Look for any identities: * 1 + π‘π‘œπ‘‘ 2 π‘₯ βˆ—
sin π‘₯ (𝑐𝑠𝑐 2 π‘₯)
sin π‘₯ (
3. Change everything to sines and cosines
4. Simplify
5. Simplify
1
sin π‘₯
1
)
𝑠𝑖𝑛2 π‘₯
csc π‘₯
Simplify: cos π‘₯ tan π‘₯ + sin π‘₯ cot π‘₯
Simplify: 1 + π‘π‘œπ‘‘ 2 π‘₯ βˆ’ π‘π‘œπ‘  2 π‘₯ βˆ’ π‘π‘œπ‘  2 π‘₯π‘π‘œπ‘‘ 2 π‘₯
THESE STEPS ARE NOT IN ANY ORDER. EACH PROBLEM IS SPECIAL
AND YOU MUST OPEN YOUR MIND TO HOW TO SOLVE THEM. Your
answers will always be 1 term, 1 number or a binomial left with sine and
cosine only 
Homework:
Page 427: #19 – 51 Odd, 57, 69
You Try It
You try It
Section 7.2
Verify Trigonometric Identities
Suggestions for Verifying trigonometric Identities
1.
2.
3.
4.
5.
Transform the more complicated side of the equation into the simplier side.
Substitute one or more basic trigonometric identity to simplify expression.
Factor or multiply to simplify
Multiply expressions by an expression equal to 1.
Express all trigonometric functions in terms of sine and cosine.
Example: Verify that 𝑠𝑒𝑐 2 π‘₯ βˆ’ tan π‘₯ cot π‘₯ = π‘‘π‘Žπ‘›2 π‘₯
1
2
𝑠𝑒𝑐 π‘₯ βˆ’ tan π‘₯
= π‘‘π‘Žπ‘›2 π‘₯
tan π‘₯
𝑠𝑒𝑐 2 π‘₯ βˆ’ 1 = π‘‘π‘Žπ‘›2 π‘₯
π‘‘π‘Žπ‘›2 π‘₯ + 1 βˆ’ 1 = π‘‘π‘Žπ‘›2 π‘₯
π‘‘π‘Žπ‘›2 π‘₯ = π‘‘π‘Žπ‘›2 π‘₯
Lesson Overview 7-2A
Lesson Overview 7-2B
Lesson Overview 7-2C
You Try It
You Try It
Section 7.3
Sum and Difference Identities
Sum and Difference Identities
Sum/Difference Identity for Sine
sin(𝛼 ± 𝛽) = sin 𝛼 cos 𝛽 ± sin 𝛽 cos 𝛼
*SINE = SIGN SAME*
Sum/Difference Identity for Cosine
cos(𝛼 ± 𝛽) = cos 𝛼 cos 𝛽 βˆ“ sin 𝛼 sin 𝛽
*COSINE = NO SIGN SAME*
Sum/Difference Identity of Tangent
tan 𝛼 ± tan 𝛽
tan(𝛼 ± 𝛽) =
1 βˆ“ tan 𝛼 tan 𝛽
*TANGENT – SAME/DIFFERENT*
Lesson Overview 7-3A
Lesson Overview 7-3B
Lesson Overview 7-3C
Examples
1. cos 105°
2. sin 165°
πœ‹
3. sin 12
4. tan
23πœ‹
12
5. sec 1275°
πœ‹
6. Find the exact value if 0 < π‘₯ < and 0 < 𝑦 <
2
3
24
cos(π‘₯ βˆ’ 𝑦) 𝑖𝑓 cos π‘₯ = π‘Žπ‘›π‘‘ cos 𝑦 =
5
25
πœ‹
2
Homework:
Page 442: #15-31 Odd, 34-38 All, 40,42
5-Minute Check Lesson 7-4A
5-Minute Check Lesson 7-4B
Lesson Overview 7-4A
Lesson Overview 7-4B
5-Minute Check Lesson 7-5A
5-Minute Check Lesson 7-5B
Lesson Overview 7-5A
5-Minute Check Lesson 7-6A
Lesson Overview 7-6A
Lesson Overview 7-6B
5-Minute Check Lesson 7-7A
5-Minute Check Lesson 7-7B
Lesson Overview 7-7A
Lesson Overview 7-7B