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Rule

Trigonometric Ratios of Notable Angles

The trigonometric ratios of some notable angles are usually known by heart.

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The above values can be calculated by either using a calculator or by using the unit circle and the Pythagorean Theorem.

Proof

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

  1. and degree angles
  2. degree angles
  3. and degree angles

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

and Degree Angles

In this section, the value of sine, cosine, and tangent of 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.

circleandangle
The terminal side of the angle and the unit circle intersect at Therefore, the cosine and sine of are and respectively. Since the tangent of an angle equals sine over cosine, the value of the tangent of is and is therefore undefined, as division by zero is not possible.
Next, the trigonometric ratios of will be calculated. The terminal side of this angle lies on the axis.
circleandangle
The terminal side of the angle and the unit circle intersect at Therefore, the cosine and sine of are and respectively. Since the tangent of an angle equals sine over cosine, the value of the tangent of is

Degree Angle

To find the trigonometric ratios of a right isosceles triangle with hypotenuse will be drawn. Since the right angle measures by the Triangle Angle Sum Theorem, the acute angles measure Let be the length of the legs of the right triangle.

isosceles triangle
By the Pythagorean Theorem, the sum of the squares of lengths of the legs is equal to the length of the hypotenuse squared.
Solve for
The length of the legs of the triangle is
isosceles triangle
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.
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 the tangent of is

and Degree Angles

Consider an equilateral triangle with a side length of

equilateral triangle

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 base length of and angles with measures and

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

Finally, the sine, cosine, and tangent of and can be obtained by using the definitions of the trigonometric ratios.

Definition Substitute Simplify

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

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Extra

Constructing the Table
The following five steps help to construct the table for notable angles.
1
Write the Notable Angles and Trigonometric Functions
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First, the heading row with the notable angles will be written. The first column containing and will also be written.


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2
Filling the Sine and Cosine Rows
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In the sine row, the integer numbers from to will be written one per column. In the cosine row, the same numbers but in the opposite order will be written.


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3
Take the Square Root of Each Number
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Now, the square root of each number will be calculated.


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4
Divide Each Number by
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Each number will now be divided by


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5
Writing the Tangent Row
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Finally, to write the third row, corresponding to the tangent ratio, the fact that will be used. The number in the sine row will be divided by the number in the cosine row.


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