Question 1 What is the distance between the polar coordinates ( 4 , 𝜋 ) (4,π) and ( 5 , 𝜋 ) (5,π)? Answer 1 Step 1: Distance formula for polar coordinates: 𝑑 = 𝑟 1 2 + 𝑟 2 2 − 2 𝑟 1 𝑟 2 cos ⁡ ( 𝜃 1 − 𝜃 2 ) d= r 1 2 +r 2 2 −2r 1 r 2 cos(θ 1 −θ 2 ) Step 2: Given points: 𝑟 1 = 4 , 𝜃 1 = 𝜋 , 𝑟 2 = 5 , 𝜃 2 = 𝜋 r 1 =4,θ 1 =π,r 2 =5,θ 2 =π Substitute into the formula: 𝑑 = 4 2 + 5 2 − 2 ( 4 ) ( 5 ) cos ⁡ ( 𝜋 − 𝜋 ) d= 4 2 +5 2 −2(4)(5)cos(π−π) 𝑑 = 16 + 25 − 2 ( 4 ) ( 5 ) ( 1 ) d= 16+25−2(4)(5)(1) 𝑑 = 41 − 40 = 1 = 1 d= 41−40 = 1 =1 Conclusion: The distance between the points is 1 1. Question 2 What is the Cartesian form of ( − 1 , 4 𝜋 3 ) (−1, 3 4π )? Answer 1 Step 1: Conversion from polar to Cartesian coordinates: 𝑥 = 𝑟 cos ⁡ 𝜃 , 𝑦 = 𝑟 sin ⁡ 𝜃 x=rcosθ,y=rsinθ Given 𝑟 = − 1 r=−1 and 𝜃 = 4 𝜋 3 θ= 3 4π : 𝑥 = − 1 ⋅ cos ⁡ ( 4 𝜋 3 ) , 𝑦 = − 1 ⋅ sin ⁡ ( 4 𝜋 3 ) x=−1⋅cos( 3 4π ),y=−1⋅sin( 3 4π ) Step 2: Values from the unit circle: cos ⁡ ( 4 𝜋 3 ) = − 1 2 , sin ⁡ ( 4 𝜋 3 ) = − 3 2 cos( 3 4π )=− 2 1 ,sin( 3 4π )=− 2 3 Substitute into the equations: 𝑥 = − 1 ⋅ ( − 1 2 ) = 1 2 , 𝑦 = − 1 ⋅ ( − 3 2 ) = 3 2 x=−1⋅(− 2 1 )= 2 1 ,y=−1⋅(− 2 3 )= 2 3 Conclusion: The Cartesian coordinates are ( 1 2 , 3 2 ) ( 2 1 , 2 3 ). Question 3 Solve for 𝑥 x in the equation 2 𝑥 2 + 15 𝑥 − 9.1 = 0 2x 2 +15x−9.1=0 using the quadratic formula. Answer 1 Step 1: Quadratic formula: 𝑥 = − 𝑏 ± 𝑏 2 − 4 𝑎 𝑐 2 𝑎 x= 2a −b± b 2 −4ac Given: 𝑎 = 2 , 𝑏 = 15 , 𝑐 = − 9.1 a=2,b=15,c=−9.1 Step 2: Calculate the discriminant: Δ = 𝑏 2 − 4 𝑎 𝑐 = 1 5 2 − 4 ( 2 ) ( − 9.1 ) = 225 + 72.8 = 297.8 Δ=b 2 −4ac=15 2 −4(2)(−9.1)=225+72.8=297.8 Step 3: Substitute into the formula: 𝑥 = − 15 ± 297.8 2 ( 2 ) = − 15 ± 297.8 4 x= 2(2) −15± 297.8 = 4 −15± 297.8 𝑥 = − 15 ± 17.26 4 x= 4 −15±17.26 Step 4: Simplify for both roots: 𝑥 1 = − 15 + 17.26 4 = 2.26 4 = 0.565 x 1 = 4 −15+17.26 = 4 2.26 =0.565 𝑥 2 = − 15 − 17.26 4 = − 32.26 4 = − 8.065 x 2 = 4 −15−17.26 = 4 −32.26 =−8.065 Conclusion: The solutions are: 𝑥 1 ≈ 0.565 , 𝑥 2 ≈ − 8.065 x 1 ≈0.565,x 2 ≈−8.065