#### Transcript unit circle

10-3 The Unit Circle Warm Up Find the measure of the reference angle for each given angle. 1. 120° 2. 225° Convert the following from degrees to radians or radians to degrees. 3. 150° Holt McDougal Algebra 2 4. 2π radians 10-3 The Unit Circle A unit circle is a circle with a radius of 1 unit. For every point P(x, y) on the unit circle, the value of r is 1. 2 2 x y 1 (0,1) (-1,0) (1,0) (0,-1) Holt McDougal Algebra 2 10-3 The Unit Circle Why the unit circle? • We use the unit circle to help us evaluate any trigonometric function. • Ex. sin 60° = • Ex. cos Π/2 = Holt McDougal Algebra 2 10-3 The Unit Circle Explore the Unit Circle • http://www.geogebratube.org/student/m26 809 • Take a few moments to go to this website and explore the unit circle. • Write down any interesting things you notice about the sine, cosine, and tangent values as you change the angle. Holt McDougal Algebra 2 10-3 The Unit Circle Creating/ completing the unit circle We will keep in mind 2 things 1. The unit circle is symmetric in many ways (reference angles) 2. The quadrant helps us to find out which is positive and which is negative. Holt McDougal Algebra 2 10-3 The Unit Circle 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 10-3 The Unit Circle Holt McDougal Algebra 2 10-3 The Unit Circle So the coordinates of P can be written as (cosθ, sinθ). tanθ = sinθ cosθ Holt McDougal Algebra 2 10-3 The Unit Circle Example 2A: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. cos 225° Holt McDougal Algebra 2 10-3 The Unit Circle Example 2B: Using the Unit Circle to Evaluate Trigonometric Functions Use the unit circle to find the exact value of each trigonometric function. tan Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 1a Use the unit circle to find the exact value of each trigonometric function. sin 315° Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 1b Use the unit circle to find the exact value of each trigonometric function. tan 180° Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 1c Use the unit circle to find the exact value of each trigonometric function. Holt McDougal Algebra 2 10-3 The Unit Circle You can use reference angles and Quadrant I of the unit circle to determine the values of trigonometric functions around the unit circle. Holt McDougal Algebra 2 10-3 The Unit Circle 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 10-3 The Unit Circle Example 3: Using Reference Angles to Evaluate Trigonometric functions Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of 330°. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3a Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of 270°. 270° Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3b Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of each angle. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3c Use a reference angle and or the unit circle to find the exact value of the sine, cosine, and tangent of each angle. –30° –30° Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3c Use a coterminal angle and the unit circle to find the exact value of the sine, cosine, and tangent of each angle. 390° Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 3c Use a coterminal angle and or the unit circle to find the exact value of the sine, cosine, and tangent of each angle. 5π/2 Holt McDougal Algebra 2 10-3 The Unit Circle Arc Length Holt McDougal Algebra 2 10-3 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. Holt McDougal Algebra 2 10-3 The Unit Circle Holt McDougal Algebra 2 10-3 The Unit Circle Example 4: Automobile Application 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? Holt McDougal Algebra 2 10-3 The Unit Circle Example 4 Continued Holt McDougal Algebra 2 10-3 The Unit Circle Example 4 Continued Step 3 Find the length of the arc intercepted by radians. Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 4 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 Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 4 Continued Holt McDougal Algebra 2 10-3 The Unit Circle Check It Out! Example 4 Continued Step 3 Find the length of the arc intercepted by radians. Holt McDougal Algebra 2 10-3 The Unit Circle Lesson Quiz: Part I Convert each measure from degrees to radians or from radians to degrees. 1. 100° 2. 3. Use the unit circle to find the exact value of 4. Use a reference angle to find the exact value of the sine, cosine, and tangent of Holt McDougal Algebra 2 . 10-3 The Unit Circle Lesson Quiz: Part II 5. A carpenter is designing a curved piece of molding for the ceiling of a museum. The curve will be an arc of a circle with a radius of 3 m. The central angle will measure 120°. To the nearest tenth of a meter, what will be the length of the molding? Holt McDougal Algebra 2