Precalculus Honors Problem Set: Elementary Trigonometry

Transcription

1 Precalculus Honors Problem Set: Elementary Trigonometry Mr. Warkentin 03 Sprague Hall Academic Year Directions: These questions are not presented in order of difficulty. Some of these questions are quick, some long. Real learning should happen when you are doing many of these homework problems - stick with questions, wrestle with them, and you will be rewarded with a much stronger understanding of our content. Unless otherwise indicated, your answers must be exact or have four significant figures. Questions labelled Challenge are genuinely tough questions that will improve your mental discipline, creativity, and persistence. While you are not required to do these questions, students considering a STEM major should at least make an attempt. Any extra credit problems in this class will be similar to challenge problems. 1. Look at the following picture. What is the length of segment C D? Adapted from a problem set published by Mr. Doherty. Look at the following picture. Use the sine, cosine, and tangent functions on your calculator to find all missing side lengths.

2 3. Construct and label a right triangle (angle measures in degrees) containing acute angle α (read: alpha) such that sin(α) = 1. What are the other trigonometric functions of α? (You do not need to find the value of α.). Now, construct and label a right triangle containing acute angle β (read: beta) such that tan(β) = 10. What are the other trigonometric functions of β? (You do not need to find the value of β.) 5. Look at the following picture. Find the perimeter of goat G s grazing area. The animal is tied to a corner of a 0 0 barn, by an 80 rope (where the symbol means distance measured in feet). Assume that there is grass everywhere except inside the barn. To solve this problem, use the properties of a triangle. 6. State 3 angles which all have the same sine value. Two of these angles must be coterminal and 1 must not. 7. A certain point on the unit circle has a y-coordinate of 0. What is that point s x-coordinate? 8. A certain point on the unit circle has an x-coordinate of 3. What is that point s y-coordinate?

3 9. Give the coordinates of 3 points on the unit circle. None of your answers should contain integer coordinates. 10. Given that cos(80 ) = , explain how to find cos(100), cos(60), cos(80), and sin(190) without using a calculator. Adapted from a problem set published by Phillips Exeter Academy 11. A wheel whose radius is 1 is placed so that its center is at (3, ). A paint spot on the rim is found at (, ). The wheel is spun θ degrees in the counterclockwise direction. Now what are the coordinates of that paint spot? Adapted from a problem set published by Phillips Exeter Academy 1. Using the unit circle and your knowledge of special right triangles, complete the following table by hand (you can only use your calculator to convert 115 to radians), then check your work with a calculator. It will help you to sketch pictures of the angles in question on the unit circle. What patterns do you notice in this table? α (in degrees) α (in radians) sin(α) cos(α) tan(α) rad rad 90 rad 0 0 rad 5 rad 115 rad 13. Look at the following picture.

4 a. Making sure it is in radian mode, use your calculator to find the sine, cosine, and tangent of α and γ. What are the relationships between these values? b. Assume that the hypotenuse of the triangle is cm. What are the lengths of the other sides? c. Drop the assumption that the hypotenuse is cm. What is another set of possible values for the lengths of this triangle s sides? 1. Draw two arcs of the same length on two different circles. Confirm that these two arcs are of the same length. 15. Look at the following picture. If each pipe is one foot in diameter, how long is the band holding them together? 16. On a circle of radius 5cm, what are the lengths of the arcs with the following central angles? a. 1 rad b c. 3 rad d Challenge: Look at the following picture. What are the lengths of segments AB, AC, and AD?

5 Adapted from a problem set published by the National Society of Professional Surveyors for the Trig-Star competition 18. From the top of Mt. Washington, which is 688 feet above sea level, how far is it to the horizon? Assume that the earth has a 3960-mile radius (one mile is 580 feet), and give your answer to the nearest mile. 19. Draw an arc such that its length (in centimeters) is greater than its radian measure. Also, draw an arc such that its length (in centimeters) is less than its radian measure. What is the differentiating factor between these two cases? 0. Look at the following picture and convert all Degree-Minute-Second (DMS) angle measures into radians, with significant figures of accuracy.

6 Adapted from a problem set published by the National Society of Professional Surveyors for the Trig-Star competition 1. Given the following pairs of angles, decide which is bigger. Try to find the answers in your head and then check with a calculator or computer. a. 1 rad vs. 5 b. rad vs c. 3 rad vs d. 11 rad vs e. f. 7 6 rad vs rad vs. 5 7 g..5 rad vs. 10 h rad vs State angles, in radians and in degrees, which are coterminal with α = 8 rad. 3. To the nearest foot, what is the length of a one-second arc on the earth s equator? Assume the radius of the earth is 3960 miles.. In a right triangle, an acute angle has the same value for its cosine and its sine. What is the measure of this acute angle in degrees? In radians? Adapted from a problem set published by Mr. Doherty 5. Textbook page 337, question: 75

7 6. Challenge: Textbook page 337, question: Given the following pairs of trigonometric functions, decide which is bigger. Try to find the answers in your head or on scratch paper and then check with a calculator or computer. a. sin 3 vs. sin b. sin 3 vs. sin c. sin (15 ) vs. sin (165 ) d. cos 3 vs. cos e. cos 3 vs. cos f. cos (15 ) vs. cos (165 ) g. sin (70 ) vs. cos (70 ) h. sin 5 vs. cos 5 i. sin ( 0 ) vs. cos ( 0 ) j. tan (80 ) vs. tan (85 ) k. tan( 85 ) vs. tan( 89 ) l. tan 3 16 vs. tan 3 15 m. tan(89 ) vs. tan(91 ) For the tangent functions, in particular, what is the super-easy way to make these comparisons? 8. Later in the year, we will prove that sin 15 8 =. Give two more angles with the same sine value. 9. Quinn is running around the circular track centered at the origin with a radius of 100 meters, at a speed of meters per second. Quinn starts at the point (100, 0) and runs in the counterclockwise direction. After 30 minutes of running, what are Quinn s coordinates? Adapted from a problem set published by Phillips Exeter Academy Honor Code In the space below, please write "I have neither given nor received unauthorized aid on this work." and sign your name.