Final answer:
Cosine (cos), tangent (tan), and sine (sin) are fundamental trigonometric functions that are defined using the ratios of sides in a right triangle. Cosine relates the adjacent side to the hypotenuse, sine relates the opposite side to the hypotenuse, and tangent relates the opposite side to the adjacent side.
Step-by-step explanation:
In the context of a right triangle, the definitions of cosine (cos), tangent (tan), and sine (sin) are based on the ratios of the sides of the triangle. If we consider a right triangle where the angle of interest is θ, the side adjacent to θ is x, the side opposite θ is y, and the hypotenuse (the longest side opposite the right angle) is h, then the formulas are as follows:
- Cosine of an angle (cos θ) is the ratio of the length of the adjacent side to the hypotenuse, hence cos θ = x/h.
- Sine of an angle (sin θ) is the ratio of the length of the opposite side to the hypotenuse, therefore sin θ = y/h.
- Tangent of an angle (tan θ) is the ratio of the length of the opposite side to the adjacent side, which means tan θ = y/x.
Additional trigonometric identities such as the law of sines (a/sin α = b/sin β = c/sin γ) and the law of cosines (c² = a² + b² - 2ab cos γ), as well as angle sum and difference formulas for sine and cosine, are vital for solving various problems involving triangles.