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2.4. EXERCISES 67

28. Using Problem 27 and the formulas for the trig functions of a sum of angles, find thefollowing. Assume x,y are small and positive if desired.

(a) cot(arccos(2x))(b) sec(arccos(x+ y))(c) csc

(arccos

(x2))

(d) cos(arcsin(x)+ arccos(y))

(e) tan(arcsin(x)+ arccos(y))

29. The function, arctan is defined as arctan(x) ≡ the angle whose tangent is x which

is in(−π

2,

π

2

). Show this is well-defined and is the inverse function for tan if the

domain of tan is restricted to be(−π

2,

π

2

). Find

(a) cos(arctan(x))(b) cot(arctan(x))(c) sin(arctan(x))

(d) csc(arctan(x))

(e) sec(arctan(x))

30. Using the formulas for the trig functions of a sum of angles, find the following.Assume x,y are small and positive if this is helpful.

(a) cot(arctan(2x))

(b) sec(arctan(x+ y))

(c) csc(arccos

(x2))

(d) cos(2arctan(x)+ arcsin(y))

31. The graphs of tan and cot suggest that these functions are periodic of period π verifythat this is indeed the case using the identities presented.

32. Give another argument which verifies the Pythagorean theorem by supplying thedetails for the following argument3. Take the given right triangle and situate copiesof it as shown below.

c ab

33. Another very simple and convincing proof of the Pythagorean theorem4 is based onwriting the area of the following trapezoid two ways. Explain why the angle denotedwith a square has radian measure equal to π/2 and find the area of the trapezoid twoways.

3This argument is old and was known to the Indian mathematician Bhaskar who lived 1114-1185 A.D.4This argument involving the area of a trapezoid is due to James Garfield 1831-1881 who was one of the

presidents of the United States. Garfield was shot early in his term as president and lingered for a couple ofmonths during which time he was attended by a physician who did not believe in the latest knowledge about theimportance of keeping wounds clean, although he was otherwise a very experienced physician who had saved thelives of many wounded men in the Civil War. It is likely that Garfield would have survived if he had receivedbetter medical care. They never found the bullet and kept probing the wound looking for it, thus introducingmore infection. If you look up Garfield, you will find many other interesting things. He was made the republicannominee by acclamation.

2.4. EXERCISES 6728.29.30.31.32.33.Using Problem 27 and the formulas for the trig functions of a sum of angles, find thefollowing. Assume x,y are small and positive if desired.(a) cot (arccos (2x)) (d) cos (arcsin (x) + arccos (y))(b) sec (arccos(x+y))(c) csc (arecos (x")) (e) tan (arcsin (x) + arccos (y))The function, arctan is defined as arctan (x) = the angle whose tangent is x whichis in (-5, >) . Show this is well-defined and is the inverse function for tan if thedomain of tan is restricted to be (-5, >) . Find(a) cos (arctan (x)) (d) csc (arctan (x))(b) cot (arctan (x))(c) sin (arctan (x)) (e) sec (arctan (x))Using the formulas for the trig functions of a sum of angles, find the following.Assume x, y are small and positive if this is helpful.(a) cot (arctan (2x)) (c) csc (arccos (x7) )(b) sec (arctan (x+y)) (d) cos (2 arctan (x) + arcsin (y))The graphs of tan and cot suggest that these functions are periodic of period 7 verifythat this is indeed the case using the identities presented.Give another argument which verifies the Pythagorean theorem by supplying thedetails for the following argument*. Take the given right triangle and situate copiesof it as shown below.c qbAnother very simple and convincing proof of the Pythagorean theorem’ is based onwriting the area of the following trapezoid two ways. Explain why the angle denotedwith a square has radian measure equal to 2/2 and find the area of the trapezoid twoways.3This argument is old and was known to the Indian mathematician Bhaskar who lived 1114-1185 A.D.4This argument involving the area of a trapezoid is due to James Garfield 1831-1881 who was one of thepresidents of the United States. Garfield was shot early in his term as president and lingered for a couple ofmonths during which time he was attended by a physician who did not believe in the latest knowledge about theimportance of keeping wounds clean, although he was otherwise a very experienced physician who had saved thelives of many wounded men in the Civil War. It is likely that Garfield would have survived if he had receivedbetter medical care. They never found the bullet and kept probing the wound looking for it, thus introducingmore infection. If you look up Garfield, you will find many other interesting things. He was made the republicannominee by acclamation.