Exercise 1. Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ.
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1 1 Radian Measures Exercise 1 Consider the following figure. The shaded portion of the circle is called the sector of the circle corresponding to the angle θ. 1. Suppose I know the radian measure of the angle θ and the arc length s. What is the radius r of the circle, in terms of s and θ?. Suppose I know the radius r and the angle θ. What is the arc length s around the fraction of the circle spanned by θ? 3. What is the area A of the shaded sector? (Hint: this part contains the fraction θ whole area of the circle.) π of the 1
2 Exercise We will use our knowledge of trigonometry to understand how a (very simplified) bicycle functions. 1. Suppose that I pedal at a rate of 10 revolutions per minute. (A revolution is one full rotation of 360 around the axis.) What is the angular velocity of the pedal system, expressed in radians per second?. The pedals are directly connected to the rear wheel, so the angular velocity of the wheels is the same as the angular speed of the pedals (which you found in part (1)). Given that the rear tire has a diameter of 69cm, what is the linear speed of a point on the rear wheel? 3. The front gear sprocket is connected to the pedals by a chain. In other words, the linear distance covered by one pedal is the same as the linear distance covered by any point on the front gear sprocket. If the length of a pedal is 6 inches and the radius of the front gear sprocket is 3 inches, what is the angular speed of the front gear sprocket?
3 Exercise 3 Use what you know about the trigonometric functions for these angles, and the relationship between degrees and radians, to complete the following table where each of the degree measures is in radians. π 3π θ 0 π 5π 5π 6 3π sin θ cos θ tan θ Trigonometric functions of real numbers Recall that for any angle θ between 0 and π, Exercise sin (θ) + cos (θ) = 1. We proved this using the pythagorean theorem and the unit circle. 1. Divide both sides of the equation above by sin (θ). What do you get? This is another important trigonometric identity.. Divide both sides of the original equation above by cos (θ). What do you get? This is another important trigonometric identity. 3
4 Exercise 5 1. Suppose t is between π and π. Draw a picture and use it to explain why cos(t) = cos( t).. Do you think cos(t) = cos( t) for any real number t? Why or why not? 3. Suppose t is between π and π. Draw a picture and use it to explain why sin(t) = sin( t).. Do you think sin(t) = sin( t) for any real number t? Why or why not? Exercise 6 1. Suppose sin t = 1 and π < t < π. Use the trigonometric identities you know, and the location of t on the unit circle, to find cos t and tan t.
5 . Suppose sec t = 3 and 0 < t < π. Use the trigonometric identities you know, and the location of t on the unit circle, to find tan and csc t. 3. Suppose t > 0 and t 1 t = 1. Solve for t using arithmetic operations.. Now, make the substitution t = cos(θ), supposing θ is between 0 and π. Solve the equation in the previous part for θ using trigonometric identities. What is cos(θ)? Are your two answers consistent? 5. Simplify the expression sin t sin t cos t cot t+1. 3 Additional Recommended Exercises , 17-, 5, 7-9, ,
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