Arkansas Tech University
MATH 1203: Trigonometry
Dr. Marcel B. Finan

11

Trigonometric Functions of Acute Angles

In this section you will learn (1) how to find the trigonometric functions using right triangles, (2) compute the values of these functions for some special angles, and (3) solve model problems involving the trigonometric functions. First, let’s review some of the features of right triangles. A triangle in which one angle is 90◦ is called a right triangle. The side opposite to the right angle is called the hypotenuse and the remaining sides are called the legs of the triangle.
Suppose that we are given an acute angle θ as shown in Figure 11.1. Note that a = 0 and b = 0.

Figure 11.1
Associated with θ are three lengths, the hypotenuse , the opposite side, and the adjacent side. We define the values of the trigonometric functions of θ as ratios of the sides of a right triangle: sin θ = csc θ =

opposite hypotenuse hypotenuse opposite =
=

b r r b cos θ = sec θ =

adjacent hypotenuse hypotenuse adjacent =
=

a r r a tan θ = cot θ =

opposite adjacent adjacent opposite =
=

b a a b √ where r = a2 + b2 (Pythagorean formula).
The symbols sin, cos, sec, csc, tan, and cot are abbreviations of sine, cosine, secant, cosecant, tangent, and cotangent. The above ratios are the same regardless of the size of the triangle. That is, the trigonometric functions defined above depend only on the angle θ. To see this, consider

1

Figure 11.2.

Figure 11.2
The triangles ∆ABC and ∆AB C are similar. Thus,
For example, using the cosine function we have cos θ =

|AB|
|AB |

=

|AC|
|AC |

=

|BC|
|B C | .

|AB|
|AB |
=
.
|AC|
|AC |

The following identities, known as reciprocal identities, follow from the definition given above. sin θ = csc θ =

1 csc θ ,
1
sin θ ,

cos θ = sec θ =

1 sec θ ,
1
cos θ ,

tan θ = cot θ =

1 cot θ ,
1
tan θ .

Example 11.1
Find the exact value