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The TheUnit UnitCircle Circle • How do we convert angle measures between degrees and radians? • How do we find the values of trigonometric functions on the unit circle? HoltMcDougal Algebra 2Algebra 2 Holt The Unit Circle You can use reference angles and Quadrant I of the unit circle to determine the values of trigonometric functions. Trigonometric Functions and Reference Angles Holt McDougal Algebra 2 The Unit Circle Students All Take Calculus The diagram shows how the signs of the trigonometric functions depend on the quadrant containing the terminal side of θ in standard position. Holt McDougal Algebra 2 The Unit Circle 0 radian 0 30 45 60 90 6 4 3 2 0 2 0 11 2 2 2 2 3 2 1 cos 1 3 2 3 3 2 2 1 2 0 sin tan 0 Holt McDougal Algebra 2 1 3 4 2 und. The Unit Circle Using Reference Angles to Evaluate Trigonometric Functions Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. Students 1. 330 All o Calculus Take 360 330 o 30 Step 1 Find the reference angle. Step 2 Find the sin, cos, and tan of the reference angle. Step 3 Adjust the signs, if needed. 1 sin 330 2 o 3 cos 330 2 3 tan 330 3 o Holt McDougal Algebra 2 o The Unit Circle Using Reference Angles to Evaluate Trigonometric Functions Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. Students All 2. 270 270° o Take Step 1 Find the reference angle. Calculus 360 270 90 o Step 2 Find the sin, cos, and tan of the reference angle. Step 3 Adjust the signs, if needed. sin 90° = 1 o cos 90° = 0 tan 90° = und. Holt McDougal Algebra 2 sin 270 1 cos 270 0 o tan 270o und. The Unit Circle Using Reference Angles to Evaluate Trigonometric Functions Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. Students All 11 3. 6 Step 1 Find the reference angle. Take Calculus 11 2 6 6 Step 2 Find the sin, cos, and tan of the reference angle. Step 3 Adjust the signs, if needed. 1 sin 11 3 11 1 6 2 cos sin 3 6 2 6 2 cos 6 2 11 3 tan 3 6 3 tan 6 3 Holt McDougal Algebra 2 The Unit Circle Using Reference Angles to Evaluate Trigonometric Functions Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle. Students 4 All 4 4. 3 Step 1 Find the reference angle. 3 Take Calculus 4 3 3 Step 2 Find the sin, cos, and tan of the reference angle. Step 3 Adjust the signs, if needed. 3 sin 4 1 4 3 3 2 cos sin 1 2 3 3 2 cos 3 2 tan 3 3 Holt McDougal Algebra 2 4 tan 3 3 The Unit Circle If you know the measure of a central angle of a circle, you can determine the length s of the arc intercepted by the angle. s r Holt McDougal Algebra 2 The Unit Circle Automobile Application 5. A tire of a car makes 653 complete rotations in 1 min. The diameter of the tire is 0.65 m. To the nearest meter, how far does the car travel in 1 s? Step 1 Find the radius of the tire. The radius is diameter. of the Step 2 Find the angle θ in radian through which the tire rotates in 1 second. 1 rotation = 2. 653 653 2 1 min 1 min 1 60 sec30 30 sec 653 s r .325 m Change to seconds 22.2 m/sec 30 sec Holt McDougal Algebra 2 The Unit Circle Automobile Application 6. An minute hand on Big Ben’s Clock Tower in London is 14 ft long. To the nearest tenth of a foot, how far does the tip of the minute hand travel in 1 minute? Step 1 Find the radius of the clock. r =14 Step 2 Find the angle θ in radian through which the hour hand rotates in 1 minute. 1 hour = 2. 2 1 hour 1 hour 60 min 30 30 min s r 14 ft 30 min Holt McDougal Algebra 2 1.5 ft/min Change to minutes The Unit Circle Lesson 10.3 Practice B Holt McDougal Algebra 2