Trigonometric Ratios of Notable Angles

This proof will be divided into three parts. Each of them will correspond to a different group of angles and their trigonometric ratios.

1. 90- and 0-degree angles
2. 45-degree angles
3. 30- and 60-degree angles

The first part will use the unit circle, while the second and third parts will use triangles.

90- and 0-Degree Angles

In this section, the value of sine, cosine, and tangent of θ= 90^(∘) will be calculated. The cosine and sine of an angle in standard position are the first and second coordinates, respectively, of the point of intersection of the terminal side of the angle and the unit circle.

The terminal side of the angle and the unit circle intersect at (0,1). Therefore, the cosine and sine of 90^(∘) are 0 and 1, respectively. Since the tangent of an angle equals sine over cosine, the value of the tangent of 90^(∘) is 1 ÷ 0 and is therefore undefined, as division by zero is not possible. sin 90^(∘)&=1 cos 90^(∘)&=0 tan 90^(∘)&=undefined Next, the trigonometric ratios of 0^(∘) will be calculated. The terminal side of this angle lies on the x-axis.

The terminal side of the angle and the unit circle intersect at (1,0). Therefore, the cosine and sine of 0^(∘) are 1 and 0, respectively. Since the tangent of an angle equals sine over cosine, the value of the tangent of 0^(∘) is 0÷ 1=0. sin 0^(∘)&=0 cos 0^(∘)&=1 tan 0^(∘)&=0

45-Degree Angle

To find the trigonometric ratios of θ=45^(∘), a right isosceles triangle with hypotenuse 1 will be drawn. Since the right angle measures 90^(∘), by the Triangle Angle Sum Theorem, the acute angles measure 45^(∘). Let x be the length of the legs of the right triangle.

By the Pythagorean Theorem, the sum of the squares of lengths of the legs is equal to the length of the hypotenuse squared. The length of the legs of the triangle is sqrt(2)2.

The sine of the acute angle of a right triangle is defined as the quotient of the lengths of the side opposite to the angle and the hypotenuse of the right triangle. Similarly, the cosine is defined as the quotient of the lengths of the side adjacent to the angle and the hypotenuse of the right triangle. cc sin 45^(∘)=sqrt(2)2/1 & cos 45^(∘)=sqrt(2)2/1 ⇓ & ⇓ sin 45^(∘)=sqrt(2)/2 & cos 45^(∘)=sqrt(2)/2 Finally, the tangent of an angle is the quotient of the sine and the cosine of the angle. Since any number divided by itself is 1, the tangent of 45^(∘) is 1. sin 45^(∘)&=sqrt(2)/2 [0.8em] cos 45^(∘)&=sqrt(2)/2 [0.8em] tan 45^(∘)&=1

30- and 60-Degree Angles

Consider an equilateral triangle with a side length of 1.

The altitude of this type of a triangle bisects the base and its opposite angle. Consider one of the right triangles obtained. This triangle has a hypotenuse length of 1, base length of 12, and angles with measures 30^(∘), 60^(∘), and 90^(∘).

The value of the altitude h can be found by using the Pythagorean Theorem. The obtained value of h can be added to the right triangle.

Finally, the sine, cosine, and tangent of 30^(∘) and 60^(∘) can be obtained by using the definitions of the trigonometric ratios.

	Definition	Substitute	Simplify
sin 30^(∘)	opp/hyp	12/1	1/2
cos 30^(∘)	adj/hyp	sqrt(3)2/1	sqrt(3)/2
tan 30^(∘)	opp/adj	12/sqrt(3)2	sqrt(3)/3
sin 60^(∘)	opp/hyp	sqrt(3)2/1	sqrt(3)/2
cos 60^(∘)	adj/hyp	12/1	1/2
tan 60^(∘)	opp/adj	sqrt(3)2/12	sqrt(3)

With the obtained results, the sine, cosine, and tangent of all five notable angles were obtained.

	θ=0^(∘)	θ=30^(∘)	θ=45^(∘)	θ=60^(∘)	θ=90^(∘)
sin θ	0	1/2	sqrt(2)/2	sqrt(3)/2	1
cos θ	1	sqrt(3)/2	sqrt(2)/2	1/2	0
tan θ	0	sqrt(3)/3	1	sqrt(3)	undefined