Name Class Date 1 E. Finding the Ratio of Arc Length to Radius

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1 Name Class Date 0- Right-Angle Trigonometry Connection: Radian Measure Essential question: What is radian, and how are radians related to degrees? CC.9 2.G.C.5 EXPLORE Finding the Ratio of Arc Length to Radius The diagram shows three circles centered at point O. The arcs between the sides of the 60 central angle are called intercepted arcs. AB is on the circle with radius unit, CD is on the circle with radius 2 units, and E C A 60 B D F Video Tutor EF is on the circle with radius units. O 2 Each intercepted arc has a different length, but because the arcs are intercepted by the same central angle, each length is the same fraction of the circumference of the circle containing the arc. A Determine the fraction of each circle s circumference that the length of each arc represents. How many degrees are in a circle? What fraction of the total number of degrees in a circle is 60? So, what fraction of the circumference of each circle is the length of each intercepted arc? B Complete the table below. To find the length of the intercepted arc, use the fraction that you found in part A. Give all answers in terms of. Radius, r 2 Circumference, C (C = 2r) Length of Intercepted Arc, s Ratio of Arc Length to Radius, s_ r REFLECT a. What do you notice about the ratios in the fourth column of the table? Chapter Lesson

2 b. When the ratio of the values of one variable y to the corresponding values of another variable x is a constant k, y is said to be proportional to x, and the constant k is called the constant of proportionality. Because _ y x = k, you can solve for y to get y = kx. In the case of arcs intercepted by a 60 central angle, is arc length s proportional to radius r? If so, what is the constant of proportionality, and what equation gives s in terms of r? c. Suppose the central angle is 90 instead of 60. Would arc length s still be proportional to radius r? If so, would the constant of proportionality still be the same? Explain. Radian Measure In the Explore and its Reflect questions, you should have reached the following conclusions:. When a central angle intercepts arcs on circles that have a common center, the ratio of each arc length s to radius r is constant. 2. When the degree of the central angle changes, the constant also changes. These facts allow you to create an alternative way of measuring angles. Instead of degree, you can use radian, defined as follows: If a central angle in a circle of radius r intercepts an arc of length s, then the angle s radian is θ = s_ r. 2 CC.9 2.A.CED.4 EXPLORE Relating Radians to Degrees Let the degree of a central angle in a circle with radius r be d, as shown. You can derive formulas that relate the angle s degree and its radian. A Find the length s of the intercepted arc XY using the verbal model below. Give the length in terms of, and simplify. Degrees in arc Length of arc = Degrees in circle Circumference of circle X r d O Y Chapter Lesson

3 B Use the result from part A to write the angle s radian θ in terms of d to find a formula for converting degrees to radians. θ = _ r s = = d C Solve the equation from part B for d to find a formula for converting radians to degrees. d = θ REFLECT 2a. What radian is equivalent to 60? Why does this make sense? Radian s are usually written in terms of, using fractions, such as 2, rather than mixed numbers. CC.9 2.N.Q. EXAMPLE Converting Between Radians and Degrees A Use the formula θ = 80 d to convert each degree to radian. Simplify the result. B Use the formula d = 80 θ to convert each radian to degree. Simplify the result. Degree Radian Radian Degree Chapter Lesson

4 REFLECT a. Which is greater, or radian? Explain. b. A radian is sometimes called a dimensionless quantity. Use unit analysis and the definition of radian to explain why this description makes sense. practice. Convert each degree to radian. Simplify the result. Degree Radian 2. Convert each radian to degree. Simplify the result. Radian Degree. When a central angle of a circle intercepts an arc whose length equals the radius of the circle, what is the angle s radian? Explain. 4. A unit circle has a radius of. What is the relationship between the radian of a central angle in a unit circle and the length of the arc that it intercepts? Explain. 5. A pizza is cut into 8 equal slices. a. What is the radian of the angle in each slice? b. If the length along the outer edge of the crust of one slice is about 7 inches, what is the diameter of the pizza to the nearest inch? (Use the formula θ = _ r s, but note that it gives you the radius of the pizza.) Chapter Lesson

5 Name Class Date 0- Additional Practice Convert each from degrees to radians or from radians to degrees Use the unit circle to find the exact value of each trigonometric function cos. tan 4 2. tan 6. sin 5 4. cos tan 60 Use a reference angle to find the exact value of the sine, cosine, and tangent of each angle Solve. 22. San Antonio, Texas, is located about 0 north of the equator. If Earth s radius is about 959 miles, approximately how many miles is San Antonio from the equator? Chapter 0 56 Lesson

6 Problem Solving The Unit Circle Gabe is spending two weeks on an archaeological dig. He finds a fragment of a circular plate that his leader thinks may be valuable. The arc length of the fragment is about the circumference of the original complete 6 plate and s.65 inches.. A similar plate found earlier has a diameter of.4 inches. Could Gabe s fragment match this plate? a. Write an expression for the radius, r, of the earlier plate. b. What is the, in radians, of a central angle, θ, that intercepts an arc that is 6 the length of the circumference of a circle? c. Write an expression for the arc length, S, intercepted by this central angle. d. How long would the arc length of a fragment be if it were 6 the circumference of the plate? e. Could Gabe s plate be a matching plate? Explain. 2. Toby finds another fragment of arc length 2.48 inches. What fraction of the outer edge of Gabe s plate would it be if this fragment were part of Gabe s plate? The diameter of a merry-go-round at the playground is 2 feet. Elijah stands on the edge and his sister pushes him around. Choose the letter for the best answer. How far does Elijah travel if he moves 4. Through what angle does Elijah move if he travels a distance of 80 feet around through an angle of radians? 4 the circumference? A 2.0 ft B 5. ft C 2.6 ft D 47. ft F 40 radians G 80 radians Virgil sets his boat on a 000-yard course keeping a constant distance from a rocky outcrop. Choose the letter for the best answer. 5. If Virgil keeps a distance of 200 yards, through what angle does he travel? A radians C 0 radians B 5 radians D 0 radians H 40 radians J 20 radians 6. If Virgil keeps a distance of 500 yards, what fraction of the circumference of a circle does he cover? F H G J 4 4 Chapter Lesson