Mathematics: Geometry - Introductory Trigonometry

Pythagorean Theorem
The side of a triangle opposite the right angle is called the hypotenuse. The other two sides are called the legs. The Pythagorean theorem states a relationship among the legs and hypotenuse of a right triangle:
are the lengths of the legs of a right triangle, and <i>c</i> is the length of the hypotenuse. Note that this formula will only work with right triangles.<br><img data-cke-saved-src=" />


Trigonometric Formulas
In the diagram below, angle C is the right angle, and side c is the hypotenuse. Side a is the side opposite to angle A and side b is the side opposite to angle
 

By Using ratios of side lengths as a means to calculate the sine, cosine, and tangent of an acute angle only works for right triangles.

Laws of Sines and Cosines
The law of sines states that
,

and C are the angles of a triangle, and a, b, and c are the sides opposite their respective angles. This formula will work with all triangles, not just right triangles.

The law of cosines is given by the formula a^2 = b^2 + c^2 – 2bc cos α

 a, b, and c are the sides of a triangle, and C is the angle opposite side c.

This is a generalized form of the Pythagorean theorem that can be used on any triangle.

P1. Calculate the following values based on triangle MNO:

(a) length of

(b)

(c) area of the triangle, if the units of the measurements are in miles
 

P1. (a) Since triangle MNO is a right triangle, we can use the simple form of Pythagoras theorem to find the missing side length:





(b) Recall that sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse.
So,

Since triangle MNO is a right triangle, we can use either of the legs as the height and the other as the base in the simple formula for the area of a triangle: