**Unformatted text preview: **, cosecant and cotangent functions as presented
in Theorem 10.11.
13. As we did in Exercise 8 in Section 10.2, let α and β be the two acute angles of a right triangle.
(Thus α and β are complementary angles.) Show that sec(α) = csc(β ) and tan(α) = cot(β ).
The fact that co-functions of complementary angles are equal in this case is not an accident
and a more general result will be given in Section 10.4.
9 You may need to review Sections 2.2 and 6.2 before attacking the next two problems. 652 Foundations of Trigonometry π
sin(θ)
< 1 for 0 < θ < . Use the diagram from
14. We wish to establish the inequality cos(θ) <
θ
2
the beginning of the section, partially reproduced below, to answer the following.
y
Q 1
P θ
O (a) Show that triangle OP B has area B (1, 0) x 1
sin(θ).
2 (b) Show that the circular sector OP B with central angle θ has area
(c) Show that triangle OQB has area 1
tan(θ).
2 (d) Comparing areas, show that sin(θ) < θ < tan(θ) for 0 < θ < 1
θ.
2 π
.
2 sin(θ)
π
< 1 for 0 < θ < .
θ
2
π
sin(θ)
for 0 < θ < . Combine th...

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