Chapter 8: Properties of Special Right Triangles & Trigonometry

Pythagorean Theorem is used to find if triangles are right triangles. If the sides of a triangle a, b, and c are such that a^2 + b^2 = c^2, then the triangle is a right triangle. If a^2 + b^2 > c^2, then the triangle is acute. If a^2 + b^2 < c^2, then the triangle is obtuse.

Some common Pythagorean sets you should know are 3 - 4 - 5, 5 - 12 - 13, 8 - 15 - 17, and 7 - 24 - 25. Any triangles with these sides, or all sides in proportion to these measures, are right triangles.

8.3 - Special Right Triangles

Thanks to GMChonk (Rachit Surve)

45 - 45 - 90 triangles have side lengths such that the legs are x and the hypotenuse is x times sqrt 2. To get a leg from the hypotenuse, divide by sqrt 2, and to get a hypotenuse from a leg, multiply by sqrt 2.

30 - 60 - 90 triangles have side lengths such that the shortest leg(opposite the 30 degree angle) is x, the longer leg (opposite the 60 degree angle) is x times sqrt 3, and the hypotenuse (opposite the right angle) is 2x.

8.4 - Trigonometry

Thanks to Ishwari Veerkar and Thomas Catuosco

SOH CAH TOA

Sine --> Opposite/ Hypotenuse

Cosine --> Adjacent/ Hypotenuse

Tangent --> Opposite/ Adjacent

On a calculator, they are spelled out as sin, cos, tan. MAKE SURE TO HAVE YOUR CALCULATOR IN DEGREE FORM!!!

The inverse is written as sin^-1, cos^-1, and tan^-1. They are also known as arcsine, arccosine, and arctangent. -------- - Nikola Birac

8.5 - Angles of Elevation & Depression

Thanks to Ishwari Veerkar

An Angle of Elevation is the angle formed by a horizontal line and an observer's line of sight to an object above the horizontal line.

An Angle of Depression is formed by a horizontal line and an observer's line of sight to an object below the horizontal line.

8.6 - Law of Sines and Cosines

Thanks to Ishwari Veerkar & Nishi Agrawal

Law of Sines: If the 3 sides namely a, b, c are opposite to the respective angles measuring A, B, C then

sin A/a = sin B/b = sin C/c

Law of Cosines: If the 3 sides namely a, b, c are opposite to the respective angles measuring A, B, C then

*can be used if you have SSS and need to find an angle or if you have SAS and are trying to find another side*

Essentially, if you have less than 2 angle measures, use this law.

a^2 = b^2 + c^2 - 2bc(cos A)

b^2 = a^2 + c^2 - 2ac(cos B)

c^2 = a^2 + b^2 -2ab(cos C)

Note: Uppercase letter variables mean angle measures while lowercase letter variables mean side lengths. And always remember that if you are trying to find an angle measure if you are given side lengths of any triangle, use the inverse trigonometric ratios (sin^-1, cos^-1, tan^-1). - Nikola Birac