Page 452 15) Sin: –0.9511; Cos: –0.3090, Tan: 3.0777 Quadrant III 17) 1) sin t: 𝟕 𝟓𝟑 𝟓𝟑 , cos t: 𝟐 𝟓𝟑 𝟓𝟑 , tan t: 𝟕 𝟐 csc t: 𝟓𝟑 𝟕 , sec t: 𝟓𝟑 𝟐 , cot t: 𝟐 𝟕 3) sin t: − 𝟔 𝟔𝟏 𝟔𝟏 , cos t: − 𝟓 𝟔𝟏 𝟔𝟏 , tan t: 𝟔 𝟓 csc t: − 𝟔𝟏 𝟔 , sec t: − 𝟔𝟏 𝟓 , cot t: 𝟓 𝟔 5) sin t: − 𝟏𝟎 𝟏𝟎𝟑 𝟏𝟎𝟑 , cos t: − 𝟑𝟎𝟗 𝟏𝟎𝟑 , tan t: − 𝟏𝟎 𝟑 𝟑 csc t: − 𝟏𝟎𝟑 𝟏𝟎 , sec t: − 𝟑𝟎𝟗 𝟑 , cot t: − 𝟑 𝟏𝟎 7) sin t: 𝟓 𝟓 , cos t: − 𝟐 𝟓 𝟓 , tan t: − 𝟏 𝟐 csc t: − 𝟓 , sec t: − 𝟓 𝟐 , cot t: −𝟐 9) sin t: − 𝟒 𝟓 , cos t: − 𝟑 𝟓 , tan t: 𝟒 𝟑 csc t: − 𝟓 𝟒 , sec t: − 𝟓 𝟑 , cot t: 𝟑 𝟒 15) Sin: –0.9511; Cos: –0.3090, Tan: 3.0777 Quadrant III 17) Sin: 0.9848; Cos: 0.1736, Tan: 5.6713 Quadrant I 19) Sin: –0.8660; Cos: –0.5, Tan: 1.7321 21) Sin: –1; Cos: 0, Tan: undefined Y-axis 23) Sin: 0.9614; Cos: –0.2752, Tan: –3.4939 Quadrant II 4/23/2019 1:22 AM 6.4B - The Unit Circle

4/23/2019 1:22 AM 6.4B - The Unit Circle

Pre-Calculus PreAP/Dual, Revised ©2014 The Unit Circle Section 6.4B Pre-Calculus PreAP/Dual, Revised ©2014 viet.dang@humble.k12.tx.us 4/23/2019 1:22 AM 6.4B - The Unit Circle

Review sin Ѳ = csc Ѳ = cos Ѳ = sec Ѳ = tan Ѳ = cot Ѳ = 4/23/2019 1:22 AM 6.4B - The Unit Circle

Reciprocal Identities Theorem sin θ= csc θ= cos θ= sec θ= tan θ= cot θ= 4/23/2019 1:22 AM 6.4B - The Unit Circle

Right Triangles – 45°, 45°, 90° 45-45-90 triangles (Drawing not to scale) Radius = 1 4/23/2019 1:22 AM 6.4B - The Unit Circle

Right Triangles – 30°, 60°, 90° 30-60-90 triangles (Drawing not to scale) Radius = 1 4/23/2019 1:22 AM 6.4B - The Unit Circle

Reminder 4/23/2019 1:22 AM 6.4B - The Unit Circle

The Unit Circle Activity 4/23/2019 1:22 AM 6.4B - The Unit Circle

The Unit Circle 4/23/2019 1:22 AM 6.4B - The Unit Circle

The Unit Circle 4/23/2019 1:22 AM 6.4B - The Unit Circle

The Unit Circle: π/4 Family cos Ѳ sin Ѳ Reference Angle: 45° π/4 45° 3π/4 135° 5π/4 225° 7π/4 315° Reference Angle: 45° π/4 45° √2/2 3π/4 135° –√2/2 5π/4 225° 7π/4 315° 4/23/2019 1:22 AM 6.4B - The Unit Circle

The Unit Circle: π/6 Family cos Ѳ sin Ѳ Reference Angle: 30° π/6 30° 5π/6 150° 7π/6 210° 11π/6 330° Reference Angle: 30° π/6 30° √3/2 1/2 5π/6 150° –√3/2 7π/6 210° –1/2 11π/6 330° 4/23/2019 1:22 AM 6.4B - The Unit Circle

The Unit Circle: π/3 Family cos Ѳ sin Ѳ Reference Angle: 60° π/3 60° 2π/3 120° 4π/3 240° 5π/3 300° Reference Angle: 60° π/3 60° 1/2 √3/2 2π/3 120° –1/2 4π/3 240° –√3/2 5π/3 300° 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 1 Solve cos 45° without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 2 Solve sin π/4 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 3 Solve cos 7π/4 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 4 Solve tan 7π/4 without a calculator 4/23/2019 1:22 AM 11-8: The Unit Circle

Your Turn Solve tan 9π/4 without a calculator 4/23/2019 1:22 AM 11-8: The Unit Circle

Example 5 Solve sin π/6 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 6 Solve tan 11π/6 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 7 Solve tan –π/3 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Your Turn Solve tan 2π/3 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 8 Solve sec 7π/4 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 9 Solve csc 4π/3 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 10 Solve cot 11π/6 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Your Turn Solve csc –11π/6 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 11 Solve cot π/2 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 12 Solve tan π/2 without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Your Turn Solve cot 4π without a calculator 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 13 Write the expression without any decimals, 𝐬𝐢𝐧 𝝅 𝟔 𝐜𝐨𝐬 𝝅 𝟐 − 𝐜𝐨𝐬 𝝅 𝟔 𝐬𝐢𝐧 𝝅 𝟐 4/23/2019 1:22 AM 6.4B - The Unit Circle

Example 14 Write the expression without any decimals, 𝐬𝐢𝐧 𝟕𝝅 𝟑 𝐜𝐨𝐬 𝟓𝝅 𝟒 + 𝐜𝐨𝐬 𝟕𝝅 𝟑 𝐬𝐢𝐧 𝟓𝝅 𝟒 4/23/2019 1:22 AM 6.4B - The Unit Circle

Your Turn Write the expression without any decimals, 𝐜𝐨𝐬 𝟑𝝅 𝟒 𝐬𝐢𝐧 𝟓𝝅 𝟔 − 𝐬𝐢𝐧 𝟕𝝅 𝟑 𝐜𝐨𝐬 𝟓𝝅 𝟔 4/23/2019 1:22 AM 6.4B - The Unit Circle

Assignment Unit Circle Worksheet 1 4/23/2019 1:22 AM 6.4B - The Unit Circle