Tri Math Worksheet

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Math 1083 Worksheet 11 Getting Ready for Solving Trigonometric Equations

Objectives: 3. Solve equations by factoring

4. Review the unit circle

5. Review right triangle trigonometry

(1) Review Solving Equations by Factoring

REVIEW: Special Identities Difference of squares: in the form ( )2 − ( )2

➢ 𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)

a) 4𝑥2 − 49 = (2𝑥)2 − 72 = (2𝑥 + 7)(2𝑥 − 7)

Perfect square trinomials: First term and last terms are perfect squares and the middle term is twice the product of first and last terms

➢ 𝑎2 + 2𝑎𝑏 + 𝑏2 = (𝑎 + 𝑏)2 ➢ 𝑎2 − 2𝑎𝑏 + 𝑏2 = (𝑎 − 𝑏)2

b) 𝑥2 + 6𝑥 + 9 = (𝑥)2 + 2(3)(𝑥) + 32 = (𝑥 + 3)2

c) 4𝑥2 − 20𝑥 + 25 = (2𝑥)2 − 2(2𝑥)(5) + 52 = (2𝑥 − 5)2

To factor 𝑎𝑥2 + 𝑏𝑥 + 𝑐 1. Factor 𝑎𝑥2 into 𝑎1𝑥 ∙ 𝑎2𝑥 and set up

parentheses

2. Find possible 𝑐1 𝑎𝑛𝑑 𝑐2 so that 𝑐 = 𝑐1 ∙ 𝑐2

3. Determine signs and check

d) 𝟐𝒙𝟐 − 𝒙 − 𝟑 (2𝑥 )(𝑥 )

3 = 1 ∙ 3 Different signs Attempt #1: (2𝑥 + 1 )(𝑥 − 3) Check: −6𝑥 + 𝑥 = −5𝑥 ≠the middle term 𝑥 Try again: Switch the numbers Attempt #2: (2𝑥 + 3 )(𝑥 − 1) Check: −2𝑥 + 3𝑥 = 𝑥 ≠ −𝑥 but we only missed the sign Try again: Change the signs Attempt #3: (2𝑥 − 3 )(𝑥 + 1)

Check: 2𝑥 − 3𝑥 = −𝑥 √

Note: You may open the link https://ggbm.at/uyFoojWM. Use the applet to review how to factor a

quadratic expression.

#1 Solve the following problems.

a) 2𝑥2 + 5𝑥 − 3 = 0 b) 2𝑥2 + 𝑥 − 6 = 0

c) 3𝑥2 + 2𝑥 + 1 = 0 d) 4𝑥2 − 3 = 0

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#2 Use the unit circle to find all values of 𝜃 on the interval 0 ≤ θ < 2π that satisfies

a) 𝑦 = 1

2

b) 𝑥 = − √3

2

c) 𝑥 = 0

d) 𝑦 = −1

#3 Find all solutions on the interval 0 ≤ θ < 2π. Give all answer in radians.

a) cos 𝑥 − 1 = 0 b) 2 sin 𝑥 + √2 = 0

REVIEW: Right triangle trigonometry Special right triangles

sin 𝜽 = 𝑶

𝑯 , tan 𝜽 =

𝑶

𝑨 , css 𝜽 =

𝑯

𝑶

cos 𝜽 = 𝑨

𝑯 , cot 𝜽 =

𝑨

𝑶 , sec 𝜽 =

𝑯

𝑨

#4 Find 𝜃 on the interval [0, 𝜋

2 )

a) tan 𝜃 = 1

√3 b) tan 𝜃 = 1 c) tan 𝜃 = 0

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REVIEW: Complete the table.

𝐬𝐢𝐧(𝒙) 𝐜𝐨𝐬(𝒙) 𝐭𝐚𝐧 (𝒙) 𝐜𝐨𝐭 (𝒙) 𝐬𝐞𝐜 (𝒙) 𝐜𝐬𝐜 (𝒙) Period

2𝜋

#5 Find all solutions. Give the general answer in radians.

a) cos 𝑥 − 1 = 0 b) 2 sin 𝑥 + √2 = 0

c) 𝑡𝑎𝑛 𝑥 − 1 = 0 d) tan 𝑥 + 1 = 0

#6. Write each using only the indicated function

a) cos2 𝑥 − 2 sin 𝑥; use sine b) cos(𝑥) − 2 sin2 𝑥; use cosine

#7 Factor each expression

a) 2 sin2 𝑥 + sin 𝑥 cos 𝑥 b) 2 cos2 𝑥 + cos 𝑥 − 1