Sin, Cos, and Tan Formulas

The trigonometric functions sine (sin), cosine (cos), and tangent (tan) are fundamental ratios used to relate the angles of a right-angled triangle to the lengths of its sides. Here’s an explanation of each function:

1. Sine (sin): The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, sin(theta) = opposite/hypotenuse.
2. Cosine (cos): The cosine of an angle in a right-angled triangle is defined as the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. Mathematically, cos(theta) = adjacent/hypotenuse.
3. Tangent (tan): The tangent of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. Mathematically, tan(theta) = opposite/adjacent.

Here’s a diagram to help visualize the relationships between the sides and angles in a right-angled triangle:

        |\
        | \
        |  \
   a    |   \   c
        |    \
        |     \
        |______\
        b    angle
          A

In this diagram, ‘a’ and ‘b’ represent the lengths of the two sides of the triangle, and ‘c’ represents the hypotenuse. Angle A is the angle of interest.

Let’s consider an example to illustrate the use of sine, cosine, and tangent:

Suppose we have a right-angled triangle with angle A measuring 30 degrees, side ‘a’ measuring 4 units, and the hypotenuse ‘c’ measuring 8 units. We can use sine, cosine, and tangent to find the remaining side lengths.

• Sine: sin(A) = opposite/hypotenuse
  sin(30°) = a/c
  sin(30°) = 4/8
  sin(30°) = 0.5
• Cosine: cos(A) = adjacent/hypotenuse
  cos(30°) = b/c
  cos(30°) = b/8
  cos(30°) = (√3/2) * 8
  cos(30°) ≈ 6.928
• Tangent: tan(A) = opposite/adjacent
  tan(30°) = a/b
  tan(30°) = 4/b
  tan(30°) ≈ 0.577

These formulas and ratios are not limited to right-angled triangles and can be used in various other applications of trigonometry as well.