1.E: Right Triangle Trigonometry Angles (Exercises)
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These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed.
1.1 Exercises
For Exercises 1-4, find the numeric value of the indicated angle(s) for the triangle
1.1.1 Find
1.1.2 Find
1.1.3 Find
1.1.4 Find
For Exercises 5-8, find the numeric value of the indicated angle(s) for the right triangle
1.1.5 Find
1.1.6 Find
1.1.7 Find
1.1.8 Find
1.1.9 A car goes 24 miles due north then 7 miles due east. What is the straight distance between the car's starting point and end point?
1.1.10 One end of a rope is attached to the top of a pole 100 ft high. If the rope is 150 ft long, what is the maximum distance along the ground from the base of the pole to where the other end can be attached? You may assume that the pole is perpendicular to the ground.
1.1.11 Prove that the hypotenuse is the longest side in every right triangle. (Hint: Is
1.1.12 Can a right triangle have sides with lengths 2, 5, and 6? Explain your answer.
1.1.13 If the lengths
(a) Show that (6,8,10) is a Pythagorean triple.
(b) Show that if (
(c) Show that (
(d) The triple in part(c) is known as Euclid's formula for generating Pythagorean triples. Write down the first ten Pythagorean triples generated by this formula, i.e. use:
1.1.14 This exercise will describe how to draw a line through any point outside a circle such that the line intersects the circle at only one point. This is called a tangent line to the circle (see the picture on the left in Figure 1.1.6), a notion which we will use throughout the text.
Figure 1.1.6
On a sheet of paper draw a circle of radius 1 inch, and call the center of that circle
1.1.15 Suppose that
1.2 Exercises
For Exercises 1-10, find the values of all six trigonometric functions of\\angles
Figure 1.2.3
1.2.1
1.2.2
1.2.3
1.2.4
1.2.5
1.2.6
1.2.7
1.2.8
1.2.9
1.2.10
For Exercises 11-18, find the values of the other five trigonometric functions of the acute angle
1.2.11
1.2.12
1.2.13
1.2.14
1.2.15
1.2.16
1.2.17
1.2.18
For Exercises 19-23, write the given number as a trigonometric function of an acute angle less than
1.2.19
1.2.20
1.2.21
1.2.22
1.2.23
For Exercises 24-28, write the given number as a trigonometric function of an acute angle greater than
1.2.24
1.2.25
1.2.26
1.2.27
1.2.28
1.2.29 In Example 1.7 we found the values of all six trigonometric functions of
(a) Does
(b) Does
(c) Does
(d) Does
1.2.30 For an acute angle
1.2.31 For an acute angle
1.2.32 For an acute angle
1.2.33 If
1.2.34 If
1.2.35 Prove the Cofunction Theorem (Theorem 1.2). (Hint: Draw a right triangle and label the angles and sides.)
1.2.36 Use Example 1.10 to find all six trigonometric functions of
1.2.37 In Figure 1.2.4,
Figure 1.2.4
(a) Find
(b) Find the length of
(c) Find the length of
(d) Figure 1.2.4 is drawn to scale. Use a protractor to measure the angle
1.2.38 In Exercise 37, verify that the area of
1.2.39 In Exercise 37, verify that the area of
1.2.40 In Exercise 37, verify that the area of
1.3 Exercises
1.3.1 From a position
1.3.2 Generalize Example 1.12: A person standing
1.3.3 As the angle of elevation from the top of a tower to the sun decreases from
1.3.4 Two banks of a river are parallel, and the distance between two points
1.3.5 A tower on one side of a river is directly east and north of points


















































1.3.6 The equatorial parallax of the moon has been observed to be approximately
the earth to the moon. (Hint: See Example 1.15.)
1.3.7 An observer on earth measures an angle of
6 and see Example 1.16.)
1.3.8 A ball bearing sits between two metal grooves, with the top groove having an angle of
1.3.9 The machine tool diagram on the right shows a symmetric worm thread, in which a circular roller of diameter
1.3.10 Repeat Exercise 9 using
1.3.11 In Exercise 9, what would the distance across the top of the worm thread have to be to make
1.3.12 For

































































































(Hint: In Figure 1.3.2 draw line segments from



















1.3.13 The machine tool diagram on the right shows a symmetric die punch. In this view, the rounded tip is part of a circle of radius




information in the diagram to find the radius

1.3.14 In the figure on the right,
For Exercises 15-23, solve the right triangle in Figure 1.3.4 using the given information.
Figure 1.3.4
1.3.15
1.3.16
1.3.17
1.3.18
1.3.19
1.3.20
1.3.21
1.3.22
1.3.23
1.3.24 In Example 1.10 in Section 1.2, we found the exact values of all six trigonometric functions of
Figure 1.3.5
Draw a semicircle of radius
This procedure can be continued indefinitely to create more semicircles. In general, it can be shown that the line segment from the center of the new semicircle to
(a) Explain why
(b) Explain why
(c) Use Figure 1.3.5 to find the exact values of
(d) Use Figure 1.3.5 to calculate the exact value of
(e) Use the same method but with an initial angle of
1.3.25 A manufacturer needs to place ten identical ball bearings against the inner side of a circular container such that each ball bearing touches two other ball bearings, as in the picture on the right. The (inner) radius of the container is
(a) Find the common radius
(b) The manufacturer needs to place a circular ring\\inside the container. What is the largest possible (outer) radius of the ring such that it is not on top\\of the ball bearings and its base is level with the\\base of the container?
1.3.26 A circle of radius
(a) Calculate the area of the octagon.
(b) If you were to increase the number of sides of the\\polygon, would the area inside it increase or decrease? What number would the area approach, if any? Explain.
(c) Inscribe a regular octagon inside the same circle. That is, draw a regular octagon such that each of its eight vertexes touches the circle. Calculate the area of this octagon.
1.3.27 The picture on the right shows a cube whose sides are of length
(a) Find the length of the diagonal line segment
(b) Find the angle
1.3.28 In Figure 1.3.6, suppose that
Figure 1.3.6
(a)
(b)
(c)
(Hint: What is the measure of the angle
1.3.29 Persons A and B are at the beach, their eyes are
1.4 Exercises
For Exercises 1-10, state in which quadrant or on which axis the given angle lies.
1.4.1
1.4.2
1.4.3
1.4.4
1.4.5
1.4.6
1.4.7
1.4.8
1.4.9
1.4.10
1.4.11 In which quadrant(s) do sine and cosine have the same sign?
1.4.12 In which quadrant(s) do sine and cosine have the opposite sign?
1.4.13 In which quadrant(s) do sine and tangent have the same sign?
1.4.14 In which quadrant(s) do sine and tangent have the opposite sign?
1.4.15 In which quadrant(s) do cosine and tangent have the same sign?
1.4.16 In which quadrant(s) do cosine and tangent have the opposite sign?
For Exercises 17-21, find the reference angle for the given angle.
1.4.17
1.4.18
1.4.19
1.4.20
1.4.21
For Exercises 22-26, find the exact values of
1.4.22
1.4.23
1.4.24
1.4.25
1.4.26
For Exercises 27-31, find the exact values of
1.4.27
1.4.28
1.4.29
1.4.30
1.4.31
For Exercises 32-36, find the exact values of
1.4.32
1.4.33
1.4.34
1.4.35
1.4.36
For Exercises 37-40, use Table 1.3 to answer the following questions.
1.4.37 Does
1.4.38 Does
1.4.39 Does
1.4.40 Does
1.4.41 Expand Table 1.3 to include all integer multiples of
1.5 Exercises
1.5.1 Let
(a) reflection of
(b) reflection of
(c) reflection of
1.5.2 Repeat Exercise 1 with
1.5.3 Repeat Exercise 1 with
1.5.4 We proved Equations 1.4-1.6 for any angle
1.5.5 Verify Equations 1.4-1.6 for
1.5.6 In Example 1.26 we used the formulas involving
For Exercises 7-14, find all angles
1.5.7
1.5.8
1.5.9
1.5.10
1.5.11
1.5.12
1.5.13
1.5.14
1.5.15 In our proof of the Pythagorean Theorem in Section 1.2, we claimed that in a right triangle
1.5.16 It can be proved without using trigonometric functions that the slopes of perpendicular lines are negative reciprocals. Let
1.5.17 Prove Equations 1.19-1.21 by using Equations 1.10-1.12 and 1.13-1.15.