Trig Lab Worksheets

44

Math 1083 WS 18 Getting Ready for Parametric Equations Objectives:

1. Solve systems of equations

2. Apply the Pythagorean identities

#1 Substitute 𝑡 = 2 − 𝑥 into 𝑦 = 𝑡2 − 4. Simplify the result.

Solving systems of equations by substitution Example: Solve {

3𝑥 + 𝑦 = 9 𝑦 − 7𝑥 = −11

Step 1: Solve one equation for one of the

variables.

Step 2: Substitute the solution expression into the

other variable.

Step 3: Solve the equation from Step 2 for the

variable.

Step 4: Substitute the solution into one

equation

Step 5: Check the proposed solution in both

equations.to solve for the other variable.

Step 1: 3𝑥 + 𝑦 = 9 ⟹ 𝑦 = 9 − 3𝑥 Step 2: Make substitution for y in equation (2)

(9 − 3𝑥) − 7𝑥 = −11 Step 3: 9 − 3𝑥 − 7𝑥 = −11

⟹ 9 − 10𝑥 = −11 ⟹ −10𝑥 = −11 − 9

⟹ −10𝑥 = −20 ⟹ 𝑥 = 2

Step 4: Plug in 𝑥 = 2 to Equation (1) 3(2) + 𝑦 = 9 ⟹ 6 + 𝑦 = 9

⟹ 𝑦 = 9 − 6 ⟹ 𝑦 = 3 Step 5: Check 3(2) + 3 = 9 true 3 − 7(2) = 3 − 14 = −11 true

#2 Solve the following the system of equations by substitution. Don’t forget to show all your working out, and to check by substitution at the end.

a) { 𝑥 − 2𝑦 = 3

𝑦 = 2𝑥 b) {

5𝑠 = 15 + 10𝑡 𝑠 − 2𝑡 = 31

45

REVIEW: The range of basic sine and cosine functions −1 ≤ sin 𝜃 ≤ 1 𝑎𝑛𝑑 − 1 ≤ cos 𝜃 ≤ 1 #3 For 0 ≤ 𝑥 < 2𝜋, find one value of 𝑥 such that 𝑓(𝑥) is the largest.

a) 𝑓(𝑥) = 5 cos 𝑥 b) 𝑓(𝑥) = −3 sin 𝑥 c) 𝑓(𝑥) = 2 cos(2𝑥) d) 𝑓(𝑥) = 1 + 2 sin 𝑥 e) 𝑓(𝑥) = 2 sin(3𝑥) f) 𝑓(𝑥) = −1 + 2 cos 𝑥 Recall: 𝐬𝐢𝐧𝟐 𝑨 + 𝐜𝐨𝐬𝟐 𝑨 = 𝟏 #4 Simplify each expression

a) Expand (sin 𝜃 + cos 𝜃)2 b) 3 cos2(2𝜃) + 3 sin2(2𝜃) − 1