In trigonometry, sine, cosine, and tangent are the three primary trigonometric functions. These functions relate the angles of a right triangle to the ratios of its sides. They are very useful for solving problems involving right triangles.

Key Points:

• In a right triangle, the three sides are:
  • Hypotenuse: The longest side, opposite the right angle.
  • Opposite: The side opposite the angle of interest.
  • Adjacent: The side next to the angle of interest (other than the hypotenuse).
• The three trigonometric functions are based on these sides.

Definitions of the Functions:

• Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
  sin(θ) = opposite / hypotenuse
• Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
  cos(θ) = adjacent / hypotenuse
• Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.
  tan(θ) = opposite / adjacent

Example 1: Using Sine

Consider a right triangle where the angle of interest is 30°, the hypotenuse is 10 units, and you want to find the length of the opposite side. Using the sine function:

• sin(30°) = opposite / hypotenuse
• sin(30°) = opposite / 10
• 0.5 = opposite / 10
• opposite = 0.5 × 10 = 5

Therefore, the length of the opposite side is 5 units.

Example 2: Using Cosine

Consider a right triangle where the angle of interest is 60°, and the hypotenuse is 12 units. You want to find the length of the adjacent side. Using the cosine function:

• cos(60°) = adjacent / hypotenuse
• cos(60°) = adjacent / 12
• 0.5 = adjacent / 12
• adjacent = 0.5 × 12 = 6

Therefore, the length of the adjacent side is 6 units.

Example 3: Using Tangent

Consider a right triangle where the angle of interest is 45°, and the length of the adjacent side is 7 units. You want to find the length of the opposite side. Using the tangent function:

• tan(45°) = opposite / adjacent
• 1 = opposite / 7
• opposite = 1 × 7 = 7

Therefore, the length of the opposite side is 7 units.

Summary:

Sine, cosine, and tangent are essential trigonometric functions used to relate the sides of a right triangle to its angles. These functions are useful for finding missing sides or angles in right triangle problems.

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