Math 367 Homework Assignment 9
due Thursday, April 28
1. A Pythagorean triple is an ordered triple (a, b, c) of positive integers such that a2 + b2 = c2 .
For example, (3, 4, 5) is a Pythagorean triple since 9+16 = 25. A Pythagorean triple (a, b, c) is
primitive if a, b, c have no factors in common. For example, (3, 4, 5) is primitive, but (6, 8, 10)
is not primitive since 6, 8, 10 have a common factor of 2.
(a) Find four distinct primitive Pythagorean triples, including (3, 4, 5). (You may find them
on the internet.)
(b) Let (a, b, c) be a Pythagorean triple, and let k be a positive integer. Prove that (ka, kb, kc)
is a Pythagorean triple. (Hint: The hypothesis is that a2 + b2 = c2 . Multiply both sides of
this equation by k 2 and rewrite. Note that (6, 8, 10) is obtained from (3, 4, 5) in this way with
k = 2.)
2. Use the Law of Cosines (Theorem 5.5.4) to prove the Converse to the Pythagorean Theorem
(Theorem 5.4.5).
◦
3. Prove, using only the
√ results of Chapter 5, that if θ is an angle of measure 30 , then
3
1
. (Hint: Use the result of In-Class Assignment 7, #2, on 30-60-90
sin θ = and cos θ =
2
2
triangles.)
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4. Prove,
√ using only the results of Chapter 5, that if θ is an angle of measure 60 , then
3
1
sin θ =
and cos θ = .
2
2
5. Prove, using only the results of Chapter 5, that if θ is an angle of measure 45◦ , then
1
1
sin θ = √ and cos θ = √ . (Hint: Use the result of Homework Assignment 8, #1, on
2
2
45-45-90 triangles.)
6. Let 4ABC be a triangle (not necessarily a right triangle), and let a = BC, b = AC, and
c = AB. In each part below, some of these side lengths and some of the angle measures are
given. Find all of the others by using the Law of Sines and the Law of Cosines, as well as the
inverse trigonometric functions on a calculator.
(a) a = 17, µ(∠A) = 45◦ , µ(∠B) = 60◦
(b) b = 5, c = 7, µ(∠A) = 30◦