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Special Triangles

In this lesson, three trigonometric ratios (sine, cosine, and tangent) will be examined in right triangles which have specific angle measurements of .
First we will review the basics. In a right triangle, one angle is 90º and the side across from this angle is called the hypotenuse. The two sides which form the 90º angle are called the legs of the right triangle. We show a right triangle below. The legs are defined as either “opposite” or “adjacent” (next to) the angle A.

We shall call the opposite side “opp,” the adjacent side “adj” and the hypotenuse “hyp.”
Definitions: In the following definitions, sine is called “sin,” cosine is called “cos” and tangent is called “tan.” The origin of these terms relates to arcs and tangents to a circle.
  1. sin(A) =
  2. cos(A) =
  3. tan(A) =
Special triangles:
  1. We examine a triangle with angles of . To do this, we first look at an equilateral triangle with all three angles 60º. A diagram is shown below.

There is also a perpendicular from one vertex. This perpendicular line gives us two triangles of . All three sides of the triangle have the same length which we will call s. The other important  segments have been labeled. We look at the right triangle on the left and see that s/2 is the side opposite the 30º angle. The hypotenuse is s. Therefore we have sin(30º) = . Using the Pythagorean Theorem we see that h = . Therefore cos(30º) = .
  1. We examine a triangle with angles of . To do this, we first look at a square with a diagonal which gives us two 45º angles and therefore a triangle to examine. A diagram is shown below.

We look at the right triangle in the upper left portion of the square with respect to its opp, adj, and hyp sides. The sides of the square are labeled s. The diagonal is labeled d. We see from the Pythagorean Theorem . This gives us . Therefore we have: sin(45º) = . Calculating the cosine will give us exactly the same answer.
Summary of values:
 sincostan
30º
45º
60º
We note also that
sin(60º) = = cos(30º)
cos(60º) = = sin(30º) 
These results follow from using the 60º angle in the
triangle along with the definitions of sine and cosine.
Let's practice:
  1. The triangle below is a triangle. What is the measure of side a?

. Therefore
  1. The triangle below is a triangle. What is the measure of side x?

We have cos(45º) = . Therefore
  1. The triangle below is a triangle. What is the measure of side x?

We have cos(45º) = Therefore
  1. An equilateral triangle has an altitude with length 15. What is a measure of a side of this triangle? A diagram is shown below.

Using the right triangle on the right side of the equilateral triangle, we see that 15 is opposite the 60º angle. Therefore sin(60º) = .

We solve for s:

Examples
Example In a triangle the hypotenuse is of length 10. What is the measure of one leg of this triangle?
What is your answer?
 
Example In a triangle, the side opposite the 60º angle has length 16. What is the length of the hypotenuse?
What is your answer?
 
Example A square has a diagonal of length 40. What is the measure of a side of this square?
What is your answer?
 
Example In a triangle the side opposite the 30º has length 6. What is the length of the other leg of this triangle?
What is your answer?
 



M Ransom

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