This document provides information about special right triangles, specifically 45-45-90 triangles and 30-60-90 triangles. It discusses the angle measurements of each type of triangle, the relationships between side lengths if the hypotenuse is defined as 1, and how to calculate the sine, cosine, and tangent values of the angles using the side lengths. Examples are provided to demonstrate calculating side lengths using the Pythagorean theorem and determining trigonometric functions for a 45-45-90 and 30-60-90 triangle.

1. Special Right
Triangles

2. 01
45-45-90 triangles

3. The measurements, and sin-
cos-tan of the angles
Right triangles are a special kind of triangle which has one angle that
is 90 degrees, and the other two angles sum up to 90 degrees to
get 180 degrees, the total number of degrees in a triangle
Within right triangles, there are 2 special types of right triangles that
have very specific (and helpful) rules and ratios. They are the 45-45-
90 triangle, and the 30-60-90 triangle. These are the breakdowns of
the degrees for both triangle types:

4. 45-45-90 triangle
Like its name suggests, a 45-45-90 triangle consists of 2 angles that are 45
degrees and the third which has 90. Because the 2 angles are congruent,
the opposite sides to both these angles have to be congruent as well. If the
hypotenuse of the 45-45-90 triangle is 1, then the other 2 sides each have a
measure of the square root of 2 divided per 2
We can use the Pythagorean Theorem to confirm this:
A2 + B2 = C2

5. 45-45-90 triangle
The hypotenuse is 1, so C2 would equal 1. If we take the square of (root
2)/2, we get 2/4 which can be simplified to ½. Since A and B are the
same, ½ + ½ = 1, meaning the lengths of A and B are accurate. Let’s try to
find the sine, cosine, and tangent of the angles that are 45 degree
For a quick recap, sine is calculated using opposite/hypotenuse,
cosine is calculated using adjacent/hypotenuse, and tangent is
calculated using opposite/adjacent

6. 45-45-90 triangle
If we take angle B and want to find the sine,
we identify the opposite side, which has a
measure of (root 2)/2, and we divide it by
the hypotenuse, which is 1, so sine would
just be (root 2)/2
The cosine would also be (root 2)/2
because the adjacent side is the same as
the opposite side. The tangent would be 1
because we divide the opposite by the
adjacent
A
B
C
45
90
45

7. 02
30-60-90 triangles

8. 30-60-90 triangle
The next type of special right triangle is a 30-60-90 triangle.
Like its name suggests, a 30-60-90 triangle consists of 1 angle that is 30
degrees, one angle that is 60 degrees, and the one which is 90
If the hypotenuse of the 30-60-90 triangle is 1, then the side opposite to
the 30 degree angle has a length of ½ and the side opposite to the 60
degree angle has a length of (root 3)/2
We can use the Pythagorean Theorem to confirm this:
A2 + B2 = c2

9. 30-60-90 triangle
The hypotenuse is 1, so C2 would equal 1. If we take the square of (root
3)/2, we get ¾. If we take the square of ½, we get ¼. 3/4 + 1/4 = 1,
meaning the lengths of A and B are accurate. Let’s try to find the sine,
cosine, and tangent of the 30 and 60 degree angles

10. 30-60-90 triangle
If we take angle A and want to find the sine, we
identify the opposite side which has a measure of
½ and we divide it by the hypotenuse which is 1,
so sine would just be 1/2. The cosine would be
(root 3)/2 because the adjacent side has a length
of
(root 3)/2
Tangent would be 1/(root 3) because you divide
½ by (root 3)/2. If we want to find the sine, cosine,
and tangent of Angle B (the 60 degree angle),
you would swap the sine and cosine values of
angle A to get sine and cosine for angle B, and
you would find the reciprocal of tan (A) to get tan
(B)
B
A
C
60
90
30

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