Detailed Summary
In this section, we focus on trigonometric ratios, which are defined as the relationships between the sides of a right triangle for given angles. Specifically, let’s consider triangle ABC, where angle A is an acute angle. The sides relative to this angle are categorized as follows:
- Opposite Side: the side opposite to angle A (BC).
- Adjacent Side: the side adjacent to angle A (AB).
- Hypotenuse: the longest side opposite the right angle (AC).
We define the trigonometric ratios for angle A as follows:
- Sine (sin A): the ratio of the length of the opposite side to the hypotenuse:
$$ ext{sin A} = \frac{BC}{AC}$$
- Cosine (cos A): the ratio of the length of the adjacent side to the hypotenuse:
$$ ext{cos A} = \frac{AB}{AC}$$
- Tangent (tan A): the ratio of the opposite side to the adjacent side:
$$ ext{tan A} = \frac{BC}{AB}$$
- Cosecant (cosec A): the reciprocal of sine:
$$ ext{cosec A} = \frac{1}{ ext{sin A}}$$
- Secant (sec A): the reciprocal of cosine:
$$ ext{sec A} = \frac{1}{ ext{cos A}}$$
- Cotangent (cot A): the reciprocal of tangent:
$$ ext{cot A} = \frac{1}{ ext{tan A}}$$
These ratios can be used to find unknown sides of triangles and are fundamental in solving real-world problems, such as calculating heights and distances based on angles of elevation or depression. The section also discusses how the definitions can extend to angles beyond acute, as well as defining trigonometric ratios for angles 0° and 90°. Overall, understanding these ratios and their interrelations enables students to solve various mathematical and practical problems.