Trigonometry/Proof of special trig values

In this book, you will learn the proofs of the simple and familiar trig values for special angles.

Proof that tan(45 degrees) = 1

Start with a square, as shown in the first image to the right. Draw the diagonal of the square by using a straight line to connect the upper right vertex with the lower left vertex, as shown in the image below.

Statement	Reason
ABCD is a square.	Given.
CB = AB	Definition of a square: the sides of a square are of equal length.
CB/AB = 1	Since AB ≠ 0, we can divide both sides by AB.
$\angle DAB=90^{\circ }$	Each angle of a square is a right angle.
$\angle CBA=90^{\circ }$	Each angle of a square is a right angle.
Triangle CBA is a right triangle.	A triangle is a right triangle if and only if it has a right angle.
$\angle CAB=45^{\circ }$	The diagonal line AC bisects right angle DAB, and half of 90o is 45o.
$\tan(45^{\circ })=CB/AB$	Trig ratios: tangent is defined as the ratio of the opposite side to the adjacent side.
$\tan(45^{\circ })=1$	Substitute equals for equals.

Proof that cos(60 degrees)=0.5

Start with a right triangle. Draw a perpendicular line from the top vertex to the midpoint of the low base, as shown in the image to the right. That line bisects the low base, BC, into two equal segments.

Statement	Reason
Triangle ABC is an equilateral triangle.	Given.
AM is a bisector of BC.	Given.
AC = BC	All three sides of an equilateral triangle are congruent to one another.
$\angle ACM=60^{\circ }$	Since the angles are equal, and since they add up to 180 degrees, each individual angle must be 60 degrees.
MC = ${\frac {1}{2}}$ BC	Definition of a bisector
MC = ${\frac {1}{2}}$ AC	Substituting equals for equals.
$\cos(60^{\circ })=MC/AC$	Trig ratios: cosine is defined as the ratio of the adjacent side to the hypotenuse.
$\cos(60^{\circ })=1/2=0.5$	Substitute equals for equals.