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Trigonometry

The Three Primary Trigonometric Comparisons

Nabil Akbarazzima Fatih

Mathematics

Introduction to the Three Primary Trigonometric Ratios

When ancient mathematicians studied triangles, they discovered useful patterns in the ratio of sides in right triangles. There are three primary trigonometric ratios that we will learn: sine (sin), cosine (cos), and tangent (tan).

sin⁡θ=opposite sidehypotenuse\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}
cos⁡θ=adjacent sidehypotenuse\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
tan⁡θ=opposite sideadjacent side\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}

Understanding the Sides of a Right Triangle

Before we go further, it's important to understand the terms used in trigonometric ratios:

  1. Hypotenuse: The longest side of a right triangle, always opposite to the right angle (90°).
  2. Opposite side: The side that is opposite to the angle θ we are examining.
  3. Adjacent side: The side that is adjacent to the angle θ we are examining (not the hypotenuse).
Visualization of Triangle Sides
Move the slider to see how the position of sides changes with the angle.
Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58
0.52 Radian
0°360°

Sine (sin θ)

The sine of angle θ is the ratio between the length of the opposite side and the length of the hypotenuse.

sin⁡θ=opposite sidehypotenuse\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}
Visualization of Sine (sin θ)
Notice how the sine value changes as the angle changes.
Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58
0.52 Radian
0°360°

Examples of Sine Values

AngleSine ValueDecimal Value
0°0°0000
30°30°12\frac{1}{2}0.50.5
45°45°22\frac{\sqrt{2}}{2}0.710.71
60°60°32\frac{\sqrt{3}}{2}0.870.87
90°90°1111

Cosine (cos θ)

The cosine of angle θ is the ratio between the length of the adjacent side and the length of the hypotenuse.

cos⁡θ=adjacent sidehypotenuse\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}
Visualization of Cosine (cos θ)
Notice how the cosine value changes as the angle changes.
Sin (60°) = 0.87Cos (60°) = 0.50Tan (60°) = 1.73
1.05 Radian
0°360°

Examples of Cosine Values

AngleCosine ValueDecimal Value
0°0°1111
30°30°32\frac{\sqrt{3}}{2}0.870.87
45°45°22\frac{\sqrt{2}}{2}0.710.71
60°60°12\frac{1}{2}0.50.5
90°90°0000

Tangent (tan θ)

The tangent of angle θ is the ratio between the length of the opposite side and the length of the adjacent side. It can also be calculated as the ratio between the sine and cosine of the same angle.

tan⁡θ=opposite sideadjacent side=sin⁡θcos⁡θ\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{\sin \theta}{\cos \theta}
Visualization of Tangent (tan θ)
Notice how the tangent value changes as the angle changes.
Sin (45°) = 0.71Cos (45°) = 0.71Tan (45°) = 1.00
0.79 Radian
0°360°

Examples of Tangent Values

AngleTangent ValueDecimal Value
0°0°0000
30°30°13\frac{1}{\sqrt{3}}0.580.58
45°45°1111
60°60°3\sqrt{3}1.731.73
90°90°UndefinedUndefined

Relationship between Sin, Cos, and Tan in the Unit Circle

To understand how these trigonometric ratios work for all angles, we can use the concept of the unit circle (a circle with radius 1).

Unit Circle and Trigonometric Ratios
Move the slider to see how the values of sin, cos, and tan change on the unit circle.
0.79 Radian
0°360°

In the unit circle:

  • The x-coordinate on the unit circle = cos θ
  • The y-coordinate on the unit circle = sin θ
  • Tan θ is the slope of the line from the center to the point on the unit circle

Relationships Between the Three Trigonometric Ratios

These three trigonometric ratios are related by the following formulas:

tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}
sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1

Exercise

Consider the following triangle with a 30° angle:

Triangle with 30° Angle
Right triangle with a 30° angle
Sin (30°) = 0.50Cos (30°) = 0.87Tan (30°) = 0.58
0.52 Radian
0°360°

If the length of the hypotenuse is 1, then:

  • The value of sin⁡30°\sin 30° = length of opposite side = 0.50.5
  • The value of cos⁡30°\cos 30° = length of adjacent side = 0.870.87
  • The value of tan⁡30°\tan 30° = sin⁡30°cos⁡30°\frac{\sin 30°}{\cos 30°} = 0.50.87\frac{0.5}{0.87} = 0.58
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Trigonometric Comparison: Sin θ and Cos θ

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Special Angles in Trigonometric Comparisons