**Isosceles and Equilateral Triangles **

**I**n this lesson, we study two special right triangles for which we are able to know both the angle measures and the trigonometric values. The first triangle is the isosceles right triangle, with two 45° angles. The other triangle has angles of measure 30°, 60°, and 90°, and is half of an equilateral triangle. So far, we have used right triangles to find sin(*x*), cos(*x*), and other trigonometric values without knowing the measure (degrees or radians) of the angle *x*. There are only a few **nice triangles** for which the measures of the angles and the measures of the sides can both be known exactly.

**The 45-45 Right Triangle**

The first of these is the isosceles right triangle. Because the hypotenuse is the longest side of a right triangle, it must be the legs that have the same length. Suppose the legs of this triangle each have a length of 1, as illustrated in Figure 9.1.

When a triangle has two sides of the same length, it will also have two angles of the same measure, as shown in Figure 9.2.

Because the angles of a triangle must add up to 180°, we calculate

*x*+

*x*+ 90° = 180°

*x*= 45°

*x*= 45° · = radians

Thus, the angle of an isosceles right triangle measures 45° or radians

The hypotenuse of this triangle can be calculated by the Pythagorean theorem:

- 1

^{2}+ 1

^{2}=

*H*

^{2}

*H*= √2

With this, as shown in Figure 9.3, we can calculate all the trigonometric values for *x* = 45° and *x* = .

**Example 1**

Find sin(45°).

sin (45°) = =

** Example 2**

Find sec().

sec() = = √2

**The 30-60 Right Triangle**

The other nice triangle is half of an equilateral triangle. If each side of an equilateral triangle has a length of 1, we have the triangle in Figure 9.4. Recall that all angles of an equilateral triangle are 60° or radians.

This is not a right triangle, so it cannot be used to calculate trigonometric values. However, if we split the triangle directly in half, we will obtain a right triangle, as depicted in Figure 9.5.

With the Pythagorean theorem, we can calculate the length O of the side opposite the 60° angle.

- + O

^{2}= 1

^{2}

- + O

^{2}= 1

- O =

We can also calculate the measure of the third angle *y*:

- 60° +

*y*+ 90° = 180°

*y*= 30°

*y*= 30° ·

With this, depicted in Figure 9.6, we can find all the trigonometric values of *x* = 60°, *x* = , *x* = 30°, and *x* = .

**Example 1**

Evaluate sin(60°).

sin(60°) =

**Example 2**

Find cot(30°).

cot(30°) =

**Example 3**

What is sec? First remember that is the same as 60°.

sec = sec(60°) = 2

Practice problems for this study guide can be found at:

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